#8096935

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	#53115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,47)(46,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(20,28)(24,31)(27,32)(29,30)(33,35,39,37)(34,40,44,42)(36,45,46,38)(41,43,48,47);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,46)(34,41,44,48)(35,45,37,38)(40,43,42,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,39,35)(34,42,44,40)(36,38,46,45)(41,47,48,43);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,38)(34,43)(35,36)(37,46)(39,45)(40,48)(41,42)(44,47);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,34)(35,40)(36,41)(37,42)(38,47)(39,44)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,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, 8004131);