#7917235

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

5103 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=ghi,e^2=i,f^2=ghi,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, 7824431);