#225467

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=de,e^b=eh,d^c=di,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132663);