#382665

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1096 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289861);