#8691005

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	#45563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^2=g,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, 8598201);