#7672194

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

4802 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)(14,15)(17,18)(19,21)(20,24)(22,26)(23,25)(27,29)(28,30)(31,32);(1,2,5,6)(3,7,4,8)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29);(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,21,25,23)(20,28,30,24)(27,31,32,29);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31);(1,4,5,3)(2,8,6,7)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)

Polycyclic presentation:

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

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, 7579390);