#8458692

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=gi,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, 8365888);