#8472917

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

4260 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=h,f^2=g,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, 8380113);