#7685324

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

4837 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=ghi,e^2=hi,f^2=gi,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, 7592520);