#410829

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1357 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23)(13,26)(14,18)(15,21)(16,24)(17,27)(20,32)(22,33)(25,31)(28,35)(29,36)(30,39)(34,40)(37,38);(1,3,4,7)(2,5,6,8)(9,11)(10,15,18,29)(12,20)(13,21)(14,22)(16,30,23,28)(17,19,26,33)(24,34)(25,35)(27,36)(31,40,37,32)(38,39);(9,12)(10,16,18,23)(11,20)(13,25)(14,24)(15,28,29,30)(17,31,26,37)(19,32,33,40)(21,35)(22,34)(27,38)(36,39);(2,6)(5,8)(12,24)(16,23)(20,34)(25,38)(28,30)(31,37)(32,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318259, #72318269, #72318313
maximal normal subgroup of	#72318259, #72318269, #72318313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318025);