#8734899

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	#43247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642095);