#8810131

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	#51132	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,27,28,24)(20,25,29,31)(21,26,30,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,7)(4,12)(5,8)(11,16)(18,23)(21,30)(24,27)(25,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(2,7)(3,10)(4,12)(13,15)(19,28)(20,29)(24,27)(25,31)(34,35)(36,43)(40,42)(44,48);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,20)(18,25)(19,26)(21,24)(22,29)(23,31)(27,30)(28,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,21)(18,26)(19,25)(20,24)(22,30)(23,32)(27,29)(28,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

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, 8717327);