#7918380

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

4872 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825576);