#8755424

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662620);