#7633184

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	#35232	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540380);