#213776

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10)(11,20,23,15)(12,24,26,17)(13,19)(14,18)(16,22,32,27)(21,29)(25,38,40,33)(28,30)(31,35)(34,36)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,21)(13,27)(14,23)(16,30)(17,29)(18,20)(19,32)(22,28)(24,34)(25,39)(26,36)(31,38)(33,37)(35,40);(9,11)(10,16)(12,21)(13,27)(14,23)(15,19)(17,31)(18,32)(20,30)(22,28)(24,37)(25,39)(26,36)(29,33)(34,38)(35,40);(9,12,14,26)(10,17,18,24)(11,21,23,36)(13,25,28,40)(15,29,20,34)(16,31,32,37)(19,33,30,38)(22,35,27,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh

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, 120972);