#7197643

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

1439 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)(17,18,22,27,23,28,37,43)(19,29,31,41,32,46,34,24)(20,33,36,48,35,25,42,40)(21,26,39,45,38,44,30,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,32)(18,24,28,41)(20,30,35,39)(21,36,38,42)(22,34,37,31)(25,26,33,44)(27,46,43,29)(40,47,48,45);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,35)(18,25,28,33)(19,30,32,39)(21,34,38,31)(22,36,37,42)(24,26,41,44)(27,40,43,48)(29,45,46,47);(3,8)(5,7)(10,11)(14,15)(17,21,37,30,23,38,22,39)(18,26,43,47,28,44,27,45)(19,20,34,42,32,35,31,36)(24,40,46,25,41,48,29,33);(9,12,13,16)(10,14,11,15)(20,35)(21,38)(25,33)(26,44)(30,39)(36,42)(40,48)(45,47)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=hi,d^2=fg,e^2=h,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, 7104839);