#7017021

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(11,13)(12,14)(15,16)(17,18,21,27,22,28,23,29)(19,30,33,42,34,24,35,41)(20,26,38,25,37,45,36,43)(31,40,48,46,32,44,47,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,33)(22,34)(23,35)(25,39)(26,44)(27,41)(28,30)(29,42)(32,37)(36,47)(38,48)(40,45)(43,46);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,27)(28,43)(29,45)(30,46)(32,34)(33,47)(35,48)(40,42)(41,44);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,32)(21,38)(22,37)(23,36)(24,40)(25,27)(28,45)(29,43)(30,44)(31,34)(33,47)(35,48)(39,42)(41,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,36,37,38)(24,41,30,42)(25,26,43,45)(31,47,32,48)(39,44,46,40)

Polycyclic presentation:

2^9|a^2=fg,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6924217);