#8095165

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(20,28)(24,29)(26,32)(30,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,47,46,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(18,22)(23,25)(24,29)(30,31)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(2,7)(3,10)(8,13)(11,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,46)(42,44);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002361);