#8096974

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

4221 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)(3,11)(4,13)(5,7)(6,9)(8,10)(12,15)(14,16)(17,18,22,26)(19,23,21,25)(20,24,28,31)(27,30,29,32)(33,34)(35,39)(36,37)(38,41)(40,45)(42,43)(44,47)(46,48);(1,3,8,7)(2,5,10,11)(4,14,6,12)(9,16,13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(1,4,8,6)(2,9,10,13)(3,12,7,14)(5,15,11,16)(17,19,22,21)(18,23,26,25)(20,27,28,29)(24,30,31,32)(33,36)(34,40)(35,42)(37,41)(38,45)(39,46)(43,47)(44,48);(1,5)(2,3)(4,15)(6,16)(7,10)(8,11)(9,12)(13,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,6,8,4)(2,9,10,13)(3,12,7,14)(5,16,11,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,38,40)(34,36,41,45)(35,43,44,46)(39,42,47,48);(1,7,8,3)(2,5,10,11)(4,14,6,12)(9,15,13,16)(17,21,22,19)(18,25,26,23)(20,27,28,29)(24,30,31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47)

Polycyclic presentation:

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

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, 8004170);