#6460771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,31,24)(20,32,35,46)(21,26,23,38)(25,34,43,36)(29,44,33,47)(30,41,37,40)(39,45,42,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,37)(22,31)(23,30)(25,42)(26,41)(27,28)(29,35)(32,44)(34,45)(36,48)(38,40)(39,43)(46,47);(10,14)(11,15)(17,20,21,34,22,35,23,36)(18,25,38,46,27,43,26,32)(19,29,37,48,31,33,30,45)(24,39,40,44,28,42,41,47);(11,15)(13,16)(17,21,22,23)(18,26,27,38)(19,30,31,37)(20,34,35,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47);(1,3,4,7)(2,6,5,8)(19,31)(20,35)(24,28)(25,43)(30,37)(32,46)(34,36)(40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367967);