#6370402

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=dfi,c^b=cfi,d^c=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, 6277598);