#8095910

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5312 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304025
maximal normal subgroup of	#72304025
maximal quotient of	#72304066, #72304086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003106);