#8445971

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=i,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, 8353167);