#8493195

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

4096 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,24)(19,25,20,30)(21,22,27,26)(28,29,31,32)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(2,7)(3,10)(9,12)(11,16)(18,24)(21,27)(25,30)(28,31)(35,44)(37,45)(40,48)(41,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,23,20)(18,25,24,30)(21,28,27,31)(22,29,26,32)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,29,26,32)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,32,30,29)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,31,30,28)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400391);