#8696721

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	#49389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,f^2=h,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, 8603917);