#7693665

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

5155 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=gh,e^2=ghi,f^2=i,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, 7600861);