#8096928

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	#45560	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(17,18)(19,26)(20,25)(21,24)(22,27)(23,28)(29,31)(30,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,23,21)(18,24,28,26)(20,29,30,22)(25,27,32,31)(36,44)(40,45)(42,48)(46,47);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,20,23,30)(18,25,28,32)(19,29,21,22)(24,27,26,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,21,23,19)(18,26,28,24)(20,22,30,29)(25,31,32,27)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,22)(18,27)(19,20)(21,30)(23,29)(24,32)(25,26)(28,31)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,24)(20,25)(21,26)(22,31)(23,28)(27,29)(30,32)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=h,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, 8004124);