#8455660

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	#36914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(34,40)(35,44)(36,46)(43,47);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(33,34)(35,41)(36,38)(37,43)(39,40)(42,46)(44,48)(45,47);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,35)(34,41)(36,45)(37,46)(38,47)(39,44)(40,48)(42,43);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,36,39,46)(34,42,40,38)(35,45,44,37)(41,43,48,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,37,39,45)(34,43,40,47)(35,46,44,36)(38,41,42,48);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,38,39,42)(34,36,40,46)(35,47,44,43)(37,41,45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=h,e^2=hi,f^2=h,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, 8362856);