#262584

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

963 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)(5,6,9,13)(7,15,18,11)(8,14,10,12)(16,20,17,19)(21,22)(23,27)(24,26)(25,28)(29,36)(30,33)(31,34)(32,35);(1,2,4,3)(5,7)(6,11)(8,17)(9,18)(10,16)(12,19)(13,15)(14,20)(22,26)(23,29)(25,31)(27,35)(30,32)(33,36);(6,12)(7,16)(8,10)(13,14)(17,18)(19,20)(21,23,24,30)(22,27,26,33)(25,32,31,29)(28,35,34,36);(1,3,4,2)(5,8,9,10)(6,12,13,14)(7,17,18,16)(11,19,15,20)(22,26)(27,33)(28,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169780);