#7208404

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207830
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,30,26)(20,31,33,36)(21,25,35,42)(24,38,40,28)(29,37,46,43)(32,47,34,48)(39,45,41,44);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,27,26)(20,28,33,38)(21,29,35,46)(24,36,40,31)(25,37,42,43)(32,44,34,45)(39,48,41,47);(2,6)(5,7)(11,15)(13,16)(17,20,30,38,22,33,19,28)(18,24,23,36,27,40,26,31)(21,34,46,44,35,32,29,45)(25,41,37,47,42,39,43,48);(3,8)(5,7)(10,14)(11,15)(17,21,22,35)(18,25,27,42)(19,29,30,46)(20,32,33,34)(23,37,26,43)(24,39,40,41)(28,44,38,45)(31,47,36,48);(2,6)(5,7)(11,15)(13,16)(21,35)(25,42)(29,46)(32,34)(37,43)(39,41)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,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, 7115600);