#7655240

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	#36259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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)(3,6)(4,8)(5,7)(9,10,13,12)(11,15,16,14)(17,18,23,28)(19,26,21,24)(20,22,27,25)(29,32,31,30);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,28,26)(20,29,27,31)(22,30,25,32);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,16,15)(17,20)(18,25)(19,29)(21,31)(22,28)(23,27)(24,32)(26,30);(1,4)(2,3)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,21,23,19)(18,26,28,24)(20,31,27,29)(22,32,25,30);(17,20)(18,22)(19,29)(21,31)(23,27)(24,30)(25,28)(26,32);(17,22)(18,27)(19,30)(20,28)(21,32)(23,25)(24,31)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562436);