#7631129

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

5238 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,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, 7538325);