#8508955

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416151);