#8095168

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	#53106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,25)(20,29)(21,26)(22,23)(24,27)(28,30)(31,32)(36,44)(40,45)(42,48)(46,47);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,24,22)(18,25,27,23)(20,28,21,31)(26,32,29,30)(34,38)(39,41)(40,45)(46,47);(2,7)(3,10)(8,13)(11,16)(17,20,24,21)(18,26,27,29)(19,28,22,31)(23,30,25,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,24,20)(18,26,27,29)(19,28,22,31)(23,32,25,30);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,22,24,19)(18,23,27,25)(20,28,21,31)(26,32,29,30)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,23)(18,19)(20,30)(21,32)(22,27)(24,25)(26,28)(29,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002364);