#8622135

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
prime radical	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

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, 8529331);