#7697272

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 7604468);