#9720536

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

5370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219217, #72219283, #72219285
maximal quotient of	#72219214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627732);