#7685718

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	#2540	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)(10,13)(14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32);(1,3,5,8)(2,6,4,7)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32);(9,10,12,13)(11,15,16,14)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28);(1,2,5,4)(3,6,8,7)(9,11,12,16)(10,14,13,15)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29);(1,4,5,2)(3,7,8,6)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31);(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=gi,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, 7592914);