#6380915

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288111);