#6380391

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×#121	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 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)(10,12)(13,14)(15,24)(16,28)(17,22)(18,23)(19,33)(20,27)(21,31)(25,39)(26,36)(29,37)(30,43)(32,35)(34,44)(38,40)(41,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,32)(18,27)(19,26)(21,34)(22,36)(23,24)(28,42)(29,44)(30,41)(31,40)(33,35)(37,38)(39,43);(13,16,17,29,18,30,19,31)(14,21,33,43,23,37,22,28)(15,25,32,44,27,41,26,40)(20,34,35,39,24,38,36,42);(6,9)(10,12)(13,17,18,19)(14,22,23,33)(15,26,27,32)(16,29,30,31)(20,35,24,36)(21,28,37,43)(25,40,41,44)(34,39,38,42);(5,7,8,11)(6,10,9,12)(15,27)(20,24)(25,41)(26,32)(34,38)(35,36)(39,42)(40,44)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=i,e^2=fi,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, 6287587);