#6370456

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×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2370 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)(13,14,17,22,18,20,28,33)(15,19,25,31,23,29,36,40)(16,21,26,34,27,32,38,42)(24,30,37,41,35,39,43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,29)(21,30)(22,31)(26,37)(27,35)(28,36)(32,39)(33,40)(34,41)(38,43)(42,44);(7,12)(9,11)(14,20)(15,23)(21,32)(22,33)(24,35)(25,36)(34,42)(37,43);(6,10)(9,11)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44);(13,16,17,26,18,27,28,38)(14,21,22,34,20,32,33,42)(15,24,25,37,23,35,36,43)(19,30,31,41,29,39,40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,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, 6277652);