#278725

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185921);