#7126988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7034184);