#210599

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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)(7,16)(8,12)(9,14)(10,15)(11,19)(13,18)(17,20)(21,22,25,29,26,24,28,33)(23,27,32,35,30,31,34,36);(1,2,3,4)(5,7,10,8)(6,12,15,16)(9,18,11,17)(13,14,20,19)(21,23)(22,27)(24,31)(25,32)(26,30)(28,34)(29,35)(33,36);(5,8,10,7)(6,13,15,20)(9,18,11,17)(12,14,16,19)(23,30)(27,31)(32,34)(35,36);(5,9,10,11)(6,14,15,19)(7,17,8,18)(12,13,16,20)(21,24,25,33,26,22,28,29)(23,31,32,36,30,27,34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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=dh,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, 117795);