#8479033

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

6011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261907
maximal normal subgroup of	#72261907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386229);