#6520474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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×#3221	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427670);