#8110756

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

4318 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017952);