#298593

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1044 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)(5,6,10,13)(7,11,16,20)(8,17,12,9)(14,19,18,15)(21,22,26,30)(23,27,33,36)(24,28,25,29)(31,34,32,35);(1,2,4,3)(5,7,10,16)(6,11,13,20)(8,18,12,14)(9,19,17,15)(21,23)(22,27)(24,32)(25,31)(26,33)(28,35)(29,34)(30,36);(5,8,9,6,10,12,17,13)(7,14,15,11,16,18,19,20)(23,31)(24,25)(27,34)(28,29)(32,33)(35,36);(1,3,4,2)(21,24,26,25)(22,28,30,29)(23,32,33,31)(27,35,36,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=f,d^2=ei,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=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205789);