#8661375

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	#54977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,15)(14,16)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,9)(4,6)(5,14)(7,11)(8,15)(10,16)(12,13)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29)(35,42)(39,46)(43,47)(44,48);(1,4,8,5)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5,8,4)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,8,14)(2,11,12,16)(3,4,15,5)(7,13,10,9)(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,9,14,13)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8568571);