#7685721

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
prime radical	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,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, 7592917);