#9061456

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

5128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265956, #72265961, #72265980
maximal normal subgroup of	#72265956, #72265961, #72265980
maximal quotient of	#72266560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968652);