#398152

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

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, 305348);