#278714

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, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=hi,d^2=ehi,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, 185910);