#8696640

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

4051 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)(3,13)(4,10)(5,15)(6,11)(7,9)(8,12)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,8,7)(2,9,12,13)(4,14,5,11)(6,15,16,10)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(35,44)(36,45)(40,46)(41,48);(1,4,8,5)(2,10,12,15)(3,14,7,11)(6,13,16,9)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,5,8,4)(2,10,12,15)(3,14,7,11)(6,9,16,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,6,8,16)(2,11,12,14)(3,15,7,10)(4,13,5,9)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,7,8,3)(2,9,12,13)(4,11,5,14)(6,15,16,10)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603836);