#8613255

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

5573 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=ghi,f^2=g,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	#72249465, #72249517, #72249526, …
maximal normal subgroup of	#72249465, #72249517, #72249526, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8520451);