#8695830

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	#51142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8603026);