#7630564

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

5012 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:

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

Polycyclic presentation:

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

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, 7537760);