#8174323

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

5054 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8081519);