#8095748

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

4059 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002944);