#8096932

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,10)(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)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(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)(35,42)(39,46)(43,47)(44,48);(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)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(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);(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)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,18)(19,24)(20,25)(21,26)(22,31)(23,28)(27,29)(30,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=h,f^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, 8004128);