#7631113

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538309);