#7678285

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585481);