#8613276

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

5185 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304012
maximal normal subgroup of	#72304012
maximal quotient of	#72304047, #72304086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8520472);