#8862313

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

5941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261948, #72262018
maximal normal subgroup of	#72261948, #72262018
maximal quotient of	#72300551

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8769509);