#8184892

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	#45518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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)(3,5,9,4)(6,14,15,10)(11,12,16,13)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3)(2,5)(4,7)(6,11)(8,9)(10,13)(12,14)(15,16)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,8,5)(2,9,7,3)(6,12,15,13)(10,16,14,11)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,5,8,4)(2,9,7,3)(6,13,15,12)(10,16,14,11)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,6,8,15)(2,10,7,14)(3,11,9,16)(4,12,5,13)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,7,8,2)(3,4,9,5)(6,14,15,10)(11,12,16,13)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092088);