#8700277

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=gh,f^2=gi,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, 8607473);