#7687141

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

5306 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)(4,7)(5,8)(9,10,13,16)(11,14,12,15)(17,18,23,25)(19,22,28,24)(20,26,29,30)(21,27,31,32);(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,21)(22,25)(23,28)(26,32)(27,30)(29,31);(1,2,5,8)(3,6,4,7)(9,11,13,12)(10,14,16,15)(18,25)(19,28)(20,29)(27,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,21)(22,27)(23,29)(24,32)(25,30)(28,31);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,21)(18,27)(19,20)(22,26)(23,31)(24,30)(25,32)(28,29);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,22)(18,28)(19,25)(20,27)(21,30)(23,24)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594337);