#391911

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 299107);