#8323153

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

4074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8230349);