#8471037

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=hi,f^2=gi,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, 8378233);