#7645914

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	#35329	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5006 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553110);