#8757107

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3777 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664303);