#8458712

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

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

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, 8365908);