#6373106

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,26,32)(16,21,27,36)(17,20,29,23)(24,31,37,43)(25,30,39,33)(28,34,40,35)(38,41,44,42);(6,9)(10,12)(13,15,18,26)(14,19,22,32)(16,24,27,37)(17,25,29,39)(20,30,23,33)(21,31,36,43)(28,38,40,44)(34,41,35,42);(5,7,8,11)(6,10,9,12)(14,20)(17,29)(19,30)(21,34)(22,23)(25,39)(28,40)(31,41)(32,33)(35,36)(38,44)(42,43);(1,3,4,2)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(25,38)(26,37)(29,40)(30,41)(32,43)(33,42)(39,44);(14,22)(16,27)(19,32)(20,23)(24,37)(28,40)(30,33)(38,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280302);