#6370294

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277490);