#7671827

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

5280 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)(3,6)(4,8)(5,7)(9,10,13,12)(11,15,16,14)(17,18,23,21)(19,27,28,24)(20,25,29,30)(22,31,32,26);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,22)(21,27)(23,28)(25,26)(29,32)(30,31);(2,7)(6,8)(18,21)(19,28)(20,29)(26,31);(1,2)(3,6)(4,8)(5,7)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,31)(28,32);(1,3,5,4)(2,6,7,8)(9,10,13,12)(11,14,16,15)(17,21,23,18)(19,27,28,24)(20,30,29,25)(22,31,32,26);(1,4,5,3)(2,8,7,6)(9,12,13,10)(11,15,16,14)(17,22)(18,26)(19,20)(21,31)(23,32)(24,25)(27,30)(28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579023);