#7829339

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

4599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,e^2=gi,f^2=g,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, 7736535);