#455416

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362612);