#8430470

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337666);