#8108363

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
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72267868
maximal subgroup of	#72267863, #72267868
maximal normal subgroup of	#72267863, #72267868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015559);