#210610

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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	C₂³	C₂×C₄×D₄
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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:

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

Polycyclic presentation:

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

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, 117806);