#7172594

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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)(17,18,23,25,21,27,22,28)(19,29,37,39,30,24,32,41)(20,33,38,44,35,26,36,42)(31,40,46,48,34,43,45,47);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19)(18,24)(20,34)(21,30)(22,32)(23,37)(25,41)(26,43)(27,29)(28,39)(31,35)(33,40)(36,46)(38,45)(42,47)(44,48);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,30)(20,35)(22,23)(24,39)(26,44)(27,28)(29,41)(33,42)(40,48)(43,47)(45,46);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20)(18,26)(19,31)(21,35)(22,36)(23,38)(24,40)(25,42)(27,33)(28,44)(29,43)(30,34)(32,45)(37,46)(39,47)(41,48);(9,11,12,15)(10,14,13,16)(18,27)(19,30)(20,35)(25,28)(32,37)(36,38)(40,43)(47,48)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=fi,d^2=fi,e^2=hi,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, 7079790);