#407986

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315182);