#6403888

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311084);