#7145493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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)(17,18,22,28)(19,30,33,24)(20,34,37,26)(21,29,23,27)(25,39,44,36)(31,41,35,45)(32,42,38,46)(40,48,43,47);(2,5)(3,7)(10,13)(11,15)(17,19,22,33)(18,24,28,30)(20,35,37,31)(21,38,23,32)(25,43,44,40)(26,45,34,41)(27,46,29,42)(36,47,39,48);(9,10)(11,14)(12,13)(15,16)(17,20,22,37)(18,25,28,44)(19,31,33,35)(21,36,23,39)(24,40,30,43)(26,27,34,29)(32,47,38,48)(41,46,45,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,39)(22,37)(23,36)(24,41)(25,29)(27,44)(28,34)(30,45)(32,48)(33,35)(38,47)(40,42)(43,46);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,29)(19,32,33,38)(20,36,37,39)(24,42,30,46)(25,34,44,26)(31,47,35,48)(40,41,43,45)

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,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052689);