#210715

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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)(5,6,10,14)(7,11,8,16)(9,18,19,13)(12,15,20,17)(21,22,26,29,25,24,28,33)(23,27,31,34,30,32,35,36);(1,2,3,4)(5,7,10,8)(6,11,14,16)(9,15,19,17)(12,18,20,13)(21,23)(22,27)(24,32)(25,30)(26,31)(28,35)(29,34)(33,36);(5,8,10,7)(6,12,14,20)(9,15,19,17)(11,18,16,13)(22,24)(23,30)(29,33)(31,35);(7,8)(11,16)(12,20)(15,17)(21,24,26,33,25,22,28,29)(23,27,31,34,30,32,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dhi,d^c=dhi,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117911);