#6378518

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,34,41)(19,23,29,37,27,35,42,46)(20,30,33,44,32,24,38,40)(28,39,43,48,31,36,45,47);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(19,27)(23,35)(28,31)(29,42)(36,39)(37,46)(43,45)(47,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19)(18,23)(20,31)(21,29)(22,27)(24,39)(25,37)(26,35)(28,32)(30,36)(33,45)(34,42)(38,43)(40,48)(41,46)(44,47);(2,5)(6,8)(19,27)(20,32)(23,35)(24,30)(29,42)(33,38)(37,46)(40,44);(17,20,21,33,22,32,34,38)(18,24,25,40,26,30,41,44)(19,28,29,43,27,31,42,45)(23,36,37,47,35,39,46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285714);