#7839912

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

4830 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,23,27)(19,24,22,26)(20,21,25,32)(28,29,30,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,28)(21,29)(22,23)(25,30)(26,27)(31,32);(1,2,4,6)(3,5,8,7)(19,22)(24,26)(28,30)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,16,15)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,15,16,14)(17,21)(18,25)(19,29)(20,27)(22,31)(23,32)(24,30)(26,28);(17,22,23,19)(18,26,27,24)(20,30,25,28)(21,31,32,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7747108);