#7697332

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

4429 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)(13,16)(14,15)(17,18,23,28)(19,26,30,24)(20,21,29,25)(22,27,31,32);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,12,15,16)(17,19)(18,24)(20,31)(21,27)(22,29)(23,30)(25,32)(26,28);(10,15)(12,16)(17,20,23,29)(18,25,28,21)(19,22,30,31)(24,27,26,32);(9,12,14,16)(10,11,15,13)(17,21,23,25)(18,20,28,29)(19,27,30,32)(22,24,31,26);(9,13,14,11)(10,16,15,12)(17,19)(18,26)(20,22)(21,27)(23,30)(24,28)(25,32)(29,31);(1,4,5,2)(3,7,8,6)(17,22)(18,27)(19,29)(20,30)(21,24)(23,31)(25,26)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604528);