#8754953

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	#51962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=h,f^2=g,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, 8662149);