#6370610

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×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 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)(7,10)(8,9)(11,12)(13,14,17,22,18,20,29,34)(15,19,25,32,23,30,37,41)(16,21,27,36,28,33,40,43)(24,31,38,42,26,35,39,44);(1,2,3,4)(6,9)(7,11)(15,23)(19,30)(24,26)(25,37)(31,35)(32,41)(38,39)(42,44);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,26)(17,25)(18,23)(20,30)(21,35)(22,32)(24,28)(27,39)(29,37)(31,33)(34,41)(36,44)(38,40)(42,43);(6,9)(10,12)(14,20)(15,23)(21,33)(22,34)(24,26)(25,37)(36,43)(38,39);(13,16,17,27,18,28,29,40)(14,21,22,36,20,33,34,43)(15,24,25,38,23,26,37,39)(19,31,32,42,30,35,41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=g,g^2=i,b^a=bf,d^a=dfi,c^b=cfi,d^c=di,e^c=ei

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, 6277806);