#7015253

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

1319 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)(17,18,21,28,22,27,39,26)(19,29,33,46,32,23,31,42)(20,34,38,44,37,25,36,24)(30,41,47,40,35,45,48,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,23,27,29)(20,35,37,30)(21,33,39,31)(24,43,44,40)(25,45,34,41)(26,46,28,42)(36,47,38,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,24,27,44)(19,30,32,35)(21,36,39,38)(23,40,29,43)(25,26,34,28)(31,47,33,48)(41,46,45,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,30)(21,38)(22,37)(23,41)(24,28)(26,44)(27,34)(29,45)(31,48)(32,35)(33,47)(36,39)(40,42)(43,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,39)(18,26,27,28)(19,31,32,33)(20,36,37,38)(23,42,29,46)(24,34,44,25)(30,47,35,48)(40,41,43,45)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6922449);