#7195271

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

1586 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)(10,13)(11,15)(17,18)(19,23)(20,31)(21,25)(22,26)(24,28)(27,30)(29,38)(32,40)(33,47)(34,37)(35,48)(36,43)(39,44)(41,45)(42,46);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,46)(24,31,40,37)(25,38,43,42)(33,45,35,44)(39,47,41,48);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,30,32,22,34,19,28)(18,24,23,37,26,40,27,31)(21,35,46,44,36,33,29,45)(25,41,38,47,43,39,42,48);(3,8)(6,7)(10,13)(14,16)(17,21)(18,25)(19,29)(20,33)(22,36)(23,38)(24,39)(26,43)(27,42)(28,44)(30,46)(31,47)(32,45)(34,35)(37,48)(40,41);(9,11,12,15)(10,14,13,16)(18,26)(21,36)(23,27)(24,40)(29,46)(31,37)(33,35)(44,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,e^2=hi,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, 7102467);