#7090760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,26,37,44)(19,28,31,40,30,45,21,24)(20,32,36,48,35,25,41,43)(29,47,34,39,33,46,38,42);(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,26,40)(20,33,35,29)(21,37,31,22)(25,42,32,39)(27,45,44,28)(34,41,38,36)(43,46,48,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,38)(22,36)(23,35)(24,39)(26,32)(27,43)(28,46)(30,33)(31,34)(37,41)(40,42)(44,48)(45,47);(11,16)(13,15)(17,21,22,19,23,31,37,30)(18,24,27,28,26,40,44,45)(20,34,36,33,35,38,41,29)(25,42,43,47,32,39,48,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,30)(20,35)(21,31)(27,44)(36,41)(39,42)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 6997956);