#249436

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#29617	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1185 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1185 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(4,11)(6,12)(7,14)(13,16)(17,18)(19,23)(20,30)(21,33)(22,26)(24,32)(25,36)(27,44)(28,42)(29,39)(31,43)(34,38)(35,41)(37,46)(40,47)(45,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,22,29)(18,23,26,39)(20,27,31,45)(21,34,36,28)(24,40,41,37)(25,42,33,38)(30,44,43,48)(32,47,35,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,43)(26,41)(28,47)(29,45)(30,33)(32,36)(34,46)(38,48)(39,40)(42,44);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(20,31)(24,41)(27,45)(30,43)(32,35)(37,40)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=i,d^2=h,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, 156632);