#8462118

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

4199 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(36,44)(41,45)(42,48)(46,47);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34,38,37)(35,40,43,39)(36,41,44,45)(42,47,48,46);(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35)(34,39)(36,42)(37,40)(38,43)(41,46)(44,48)(45,47);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,36,38,44)(34,41,37,45)(35,42,43,48)(39,46,40,47);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,38,34)(35,40,43,39)(36,45,44,41)(42,47,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8369314);