#8932207

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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:

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

Polycyclic presentation:

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

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, 8839403);