#10451668

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

5434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333952, #72336977
maximal normal subgroup of	#72333952, #72336977
maximal quotient of	#72334034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358864);