#8183735

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

4135 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090931);