#8734172

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

4156 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34,39,38)(35,44,46,40)(36,42,47,41)(37,48,45,43);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(19,26)(22,25)(27,28)(31,32)(33,35)(34,40)(36,37)(38,44)(39,46)(41,48)(42,43)(45,47);(3,11)(4,12)(7,16)(8,13)(17,19,21,26)(18,22,24,25)(20,27,30,28)(23,31,29,32)(33,36,39,47)(34,41,38,42)(35,37,46,45)(40,48,44,43);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,36)(34,42)(35,45)(37,46)(38,41)(39,47)(40,48)(43,44);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,37,39,45)(34,43,38,48)(35,36,46,47)(40,42,44,41);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,29)(25,32)(26,28)(33,38,39,34)(35,44,46,40)(36,42,47,41)(37,43,45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8641368);