#8075418

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

4801 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)(3,7)(10,13)(14,16)(17,18)(19,23)(20,25)(21,31)(22,24)(26,28)(27,32)(29,30);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,32)(25,26)(30,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(21,28)(25,30);(3,7)(4,8)(17,20,22,29)(18,25,24,30)(19,28,27,21)(23,26,32,31);(1,3)(2,4)(5,7)(6,8)(17,21,22,28)(18,26,24,31)(19,29,27,20)(23,25,32,30);(1,4)(2,3)(5,8)(6,7)(17,18)(19,23)(20,30)(21,26)(22,24)(25,29)(27,32)(28,31)

Polycyclic presentation:

2^9|c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982614);