#249453

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1047 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,17,45)(4,18,22,53)(5,24,28,40)(6,11,29,43)(8,30,33,16)(9,34,38,21)(10,26,42,50)(14,37,47,36)(15,41,27,32)(19,54,57,64)(20,52,23,31)(25,51,55,63)(35,60,59,46)(39,56,62,48)(44,49,58,61);(1,3,7,17)(2,8,12,33)(4,14,22,47)(5,15,28,27)(6,16,29,30)(9,18,38,53)(10,32,42,41)(11,13,43,45)(19,46,57,60)(20,34,23,21)(24,58,40,44)(25,48,55,56)(26,49,50,61)(31,36,52,37)(35,51,59,63)(39,54,62,64);(1,4,6,23)(2,9,11,37)(3,14,16,21)(5,25,26,19)(7,22,29,20)(8,31,13,18)(10,39,40,35)(12,38,43,36)(15,48,49,46)(17,47,30,34)(24,59,42,62)(27,56,61,60)(28,55,50,57)(32,51,44,54)(33,52,45,53)(41,63,58,64);(1,5,6,26)(2,10,11,40)(3,15,16,49)(4,19,23,25)(7,28,29,50)(8,32,13,44)(9,35,37,39)(12,42,43,24)(14,46,21,48)(17,27,30,61)(18,51,31,54)(20,55,22,57)(33,41,45,58)(34,56,47,60)(36,62,38,59)(52,64,53,63)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156649);