#6389058

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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

1971 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)(8,11)(9,12)(13,14,17,19)(15,23,25,36)(16,21,27,33)(18,30,29,22)(20,31,32,26)(24,37,38,44)(28,41,40,35)(34,42,43,39);(1,3,4,2)(5,7,10,6)(8,11,9,12)(14,19)(20,32)(21,33)(22,30)(23,36)(34,43)(35,41)(37,44);(13,15,17,25)(14,20,19,32)(16,24,27,38)(18,31,29,26)(21,34,33,43)(22,36,30,23)(28,42,40,39)(35,44,41,37);(5,8,10,9)(6,11,7,12)(13,16,17,27)(14,21,19,33)(15,24,25,38)(18,28,29,40)(20,34,32,43)(22,35,30,41)(23,37,36,44)(26,39,31,42);(5,9,10,8)(6,11,7,12)(14,19)(16,27)(20,32)(22,30)(23,36)(24,38)(28,40)(39,42)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,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, 6296254);