#8696775

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	#59528	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(18,20)(19,25)(24,28)(27,29)(33,34,40,39)(35,38,44,43)(36,45,37,41)(42,48,46,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,35)(34,38)(36,46)(37,42)(39,43)(40,44)(41,48)(45,47);(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36,40,37)(34,41,39,45)(35,42,44,46)(38,47,43,48);(1,4,7,2)(3,8,5,10)(6,9,14,12)(11,16,13,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37,40,36)(34,41,39,45)(35,42,44,46)(38,48,43,47);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,20,22,18)(19,26,25,23)(21,24,30,28)(27,31,29,32)(33,38)(34,35)(36,47)(37,48)(39,44)(40,43)(41,42)(45,46);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,34)(35,43,44,38)(36,45,37,41)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603971);