#8618108

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	#41087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525304);