#8094595

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

4115 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)(3,9)(4,13)(5,10)(6,7)(8,11)(12,14)(15,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(36,44)(40,45)(42,48)(46,47);(1,3,8,6)(2,9,11,7)(4,12,5,15)(10,16,13,14)(17,19,22,27)(18,20,24,25)(21,26,30,32)(23,28,29,31)(34,38)(39,41)(40,45)(46,47);(1,4,8,5)(2,10,11,13)(3,12,6,15)(7,14,9,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,5,8,4)(2,10,11,13)(3,12,6,15)(7,16,9,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,6,8,3)(2,7,11,9)(4,12,5,15)(10,16,13,14)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,7)(2,3)(4,14)(5,16)(6,11)(8,9)(10,12)(13,15)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

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, 8001791);