#8155032

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

4175 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)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34,38,42)(35,41,37,39)(36,40,46,44)(43,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(2,6)(7,9)(8,13)(14,15)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

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

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, 8062228);