#10320627

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

5272 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)(11,16)(12,15)(13,14)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30);(1,2)(3,7)(4,5)(6,8)(9,11,14,12)(10,15,13,16)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27);(2,5)(6,7)(9,10,14,13)(11,15,12,16)(18,23)(19,25)(24,29)(26,30);(2,5)(6,7)(9,10,14,13)(11,16,12,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,15,12,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300627
maximal normal subgroup of	#72300627
maximal quotient of	#72300642, #72300837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227823);