#6399803

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,19,25,33)(16,27,28,21)(17,20,30,22)(24,37,39,32)(26,31,40,34)(29,36,41,35)(38,43,44,42);(6,9)(7,11)(13,15,18,25)(14,19,23,33)(16,24,28,39)(17,26,30,40)(20,31,22,34)(21,32,27,37)(29,38,41,44)(35,42,36,43);(14,20)(17,30)(19,31)(21,35)(22,23)(26,40)(27,36)(29,41)(32,42)(33,34)(37,43)(38,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,36)(23,27)(25,39)(26,38)(30,41)(31,42)(33,37)(34,43)(40,44);(1,3,4,2)(6,9)(10,12)(15,25)(16,28)(19,33)(21,27)(26,40)(29,41)(31,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306999);