#8096899

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=gh,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, 8004095);