#7693765

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

4882 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gh,e^2=gh,f^2=h,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, 7600961);