#114287

order	512
nilpotency class	3
isoclinism	isoclinic class of #114282
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21483);