#8174303

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

5052 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081499);