#8456843

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	#43893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5034 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8364039);