#6380290

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(6,9)(10,11)(13,14)(15,24)(16,28)(17,22)(18,23)(19,33)(20,26)(21,31)(25,39)(27,36)(29,37)(30,43)(32,35)(34,44)(38,41)(40,42);(5,6,8,9)(7,11,12,10)(13,15,18,26)(14,20,23,24)(16,25,30,40)(17,32,19,27)(21,34,37,38)(22,36,33,35)(28,42,43,39)(29,44,31,41);(13,16,19,31,18,30,17,29)(14,21,22,28,23,37,33,43)(15,25,27,41,26,40,32,44)(20,34,36,42,24,38,35,39);(5,7,8,12)(6,10,9,11)(15,26)(20,24)(25,40)(27,32)(34,38)(35,36)(39,42)(41,44);(13,17,18,19)(14,22,23,33)(15,27,26,32)(16,29,30,31)(20,35,24,36)(21,28,37,43)(25,41,40,44)(34,39,38,42)

Polycyclic presentation:

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

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, 6287486);