#6473316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2400 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei

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, 6380512);