#7866496

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

4737 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7773692);