#8097321

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	#45524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28)(33,34)(35,38)(36,45)(37,47)(39,42)(40,43)(41,48)(44,46);(4,12)(7,14)(10,16)(13,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35)(34,36)(37,39)(38,43)(40,45)(41,42)(44,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,23,21)(19,20,24,26)(22,25,30,29)(27,28,31,32)(33,36,40,38)(34,35,43,45)(37,46,48,42)(39,41,44,47);(1,4,5,12)(2,7,9,14)(3,10,11,16)(6,13,8,15)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,37)(34,41)(35,44)(36,46)(38,42)(39,45)(40,48)(43,47);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,23,18)(19,26,24,20)(22,25,30,29)(27,28,31,32)(33,38,40,36)(34,35,43,45)(37,42,48,46)(39,41,44,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,39,40,44)(34,42,43,46)(35,37,45,48)(36,41,38,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004517);