#6371596

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278792);