#415234

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

664 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322430);