#7046376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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)(10,11)(12,16)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,32)(29,41)(30,42)(33,47)(34,38)(35,39)(36,48)(37,44)(40,46)(43,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,37,30)(25,31,41,38)(26,42,44,39)(33,45,36,46)(40,48,43,47);(9,11,13,10)(12,15,16,14)(17,20,19,29,22,34,23,32)(18,25,24,38,27,41,28,31)(21,36,30,46,37,33,35,45)(26,43,39,47,44,40,42,48);(3,8)(6,7)(10,11)(14,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,35)(24,39)(25,40)(27,44)(28,42)(29,45)(31,47)(32,46)(34,36)(38,48)(41,43);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(18,27)(21,37)(24,28)(25,41)(30,35)(31,38)(33,36)(45,46)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=gi,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^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6953572);