#6396527

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1876 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)(6,9)(7,11)(13,14,17,19)(15,23,26,36)(16,21,28,32)(18,30,24,22)(20,33,31,27)(25,37,39,44)(29,41,38,35)(34,43,42,40);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,19)(16,28)(20,31)(22,30)(23,36)(25,39)(29,38)(40,43);(1,3,4,2)(13,15,24,33,17,26,18,27)(14,20,30,36,19,31,22,23)(16,25,38,43,28,39,29,40)(21,34,41,44,32,42,35,37);(6,9)(10,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,32)(20,34)(22,35)(23,37)(24,38)(26,39)(27,40)(30,41)(31,42)(33,43)(36,44);(5,7,8,11)(6,10,9,12)(14,19)(16,28)(20,31)(22,30)(23,36)(25,39)(29,38)(40,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303723);