#8797760

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

5095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,6)(3,5,7,8)(18,20)(19,22)(25,28)(26,29);(3,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,18,23,20)(19,24,22,27)(21,28,30,25)(26,32,29,31);(1,2)(3,5)(4,6)(7,8)(10,15)(11,12)(17,19,23,22)(18,24,20,27)(21,29,30,26)(25,32,28,31);(9,11,14,12)(10,16,15,13)(17,20,23,18)(19,24,22,27)(21,28,30,25)(26,31,29,32);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,13,15,16)(17,21,23,30)(18,25,20,28)(19,26,22,29)(24,31,27,32);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,22,23,19)(18,24,20,27)(21,29,30,26)(25,31,28,32)

Polycyclic presentation:

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

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, 8704956);