#379987

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1438 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:

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

Polycyclic presentation:

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

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, 287183);