#6384192

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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

2086 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)(10,12)(13,14,19,23)(15,20,26,35)(16,28,30,36)(17,22,18,31)(21,32,37,29)(24,38,40,43)(25,34,27,41)(33,42,44,39);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,38)(29,39)(30,40)(31,41)(32,42)(36,43)(37,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,29)(18,32)(19,30)(20,33)(22,36)(23,37)(25,39)(26,40)(27,42)(28,31)(34,43)(35,44)(38,41);(6,9)(10,12)(13,17,19,18)(14,22,23,31)(15,25,26,27)(16,29,30,32)(20,34,35,41)(21,36,37,28)(24,39,40,42)(33,43,44,38);(1,3,4,2)(15,26)(20,35)(24,40)(25,27)(33,44)(34,41)(38,43)(39,42)

Polycyclic presentation:

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

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, 6291388);