#7197531

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 7104727);