#8475182

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

4118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382378);