#8599022

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

5653 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)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(2,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(3,7)(6,8)(9,10,12,14)(11,13,15,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,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	#72261791, #72261907
maximal normal subgroup of	#72261791, #72261907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506218);