#7648400

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	#34937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4788 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)(3,5)(4,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,22,26)(19,23,25,21)(20,24,30,28)(27,31,32,29);(2,3)(6,7)(9,11,13,16)(10,14,12,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29);(1,3,5,2)(4,7,8,6)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32);(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,f^2=hi,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, 7555596);