#6451873

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303310, #72303511, #72303523, …
maximal normal subgroup of	#72303310, #72303511, #72303523, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359069);