#8095026

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=i,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, 8002222);