#8504743

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	#37171	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,10)(4,9)(5,14)(6,7)(8,13)(11,12)(15,16)(17,18,22,26)(19,25,29,23)(20,30,21,24)(27,31,28,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,15)(14,16)(17,19,22,29)(18,23,26,25)(20,27,21,28)(24,31,30,32)(34,39)(40,43)(41,42)(47,48);(1,4,7,14)(2,9,6,5)(3,11,13,16)(8,15,10,12)(17,20,22,21)(18,24,26,30)(19,27,29,28)(23,31,25,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,5)(2,4)(3,12)(6,14)(7,9)(8,11)(10,16)(13,15)(17,18)(19,25)(20,24)(21,30)(22,26)(23,29)(27,32)(28,31)(33,35,38,46)(34,40,39,43)(36,44,37,45)(41,47,42,48);(1,6,7,2)(3,8,13,10)(4,5,14,9)(11,15,16,12)(17,19)(18,25)(20,27)(21,28)(22,29)(23,26)(24,32)(30,31)(33,36,38,37)(34,41,39,42)(35,44,46,45)(40,47,43,48);(1,3)(2,10)(4,11)(5,15)(6,8)(7,13)(9,12)(14,16)(17,21,22,20)(18,24,26,30)(19,28,29,27)(23,31,25,32)(33,37,38,36)(34,42,39,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411939);