#8661378

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	#56013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8568574);