#418595

order	512
nilpotency class	3
isoclinism	isoclinic class of #417468
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325791);