#298355

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45996	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1193 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,14)(8,18)(9,12)(10,13)(11,17)(15,16)(19,20)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,35)(34,36);(1,2,4,3)(5,6)(7,15)(8,12)(9,18)(10,13)(11,17)(14,16)(19,20)(23,30)(24,33)(27,32)(29,35);(5,7,8,17,10,16,9,19)(6,11,12,14,13,20,18,15)(21,23,24,31,25,30,33,36)(22,26,27,29,28,34,32,35);(1,3,4,2)(7,16)(11,20)(14,15)(17,19)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205551);