#6433528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,40)(20,25,31,37)(22,34,33,26)(24,35,36,30)(28,41,42,48)(32,45,44,39)(38,46,47,43);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,36)(25,37)(26,34)(27,40)(38,47)(39,45)(41,48);(11,15)(13,16)(17,19,21,29)(18,24,23,36)(20,28,31,42)(22,35,33,30)(25,38,37,47)(26,40,34,27)(32,46,44,43)(39,48,45,41);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,21,31)(18,25,23,37)(19,28,29,42)(22,32,33,44)(24,38,36,47)(26,39,34,45)(27,41,40,48)(30,43,35,46);(20,31)(25,37)(28,42)(32,44)(38,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340724);