#6493018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(11,15)(14,16)(17,18,24,30)(19,25,33,42)(20,34,21,27)(22,29,23,28)(26,37,44,36)(31,40,35,48)(32,45,38,41)(39,46,43,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,30,42)(20,35,21,31)(22,38,23,32)(26,43,44,39)(27,40,34,48)(28,45,29,41)(36,46,37,47);(10,13)(14,16)(17,20,24,21)(18,26,30,44)(19,31,33,35)(22,36,23,37)(25,39,42,43)(27,28,34,29)(32,46,38,47)(40,41,48,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,24,20)(18,27,30,34)(19,31,33,35)(22,36,23,37)(25,40,42,48)(26,28,44,29)(32,47,38,46)(39,41,43,45);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,32,33,38)(20,36,21,37)(25,41,42,45)(26,34,44,27)(31,46,35,47)(39,48,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6400214);