#8657955

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	#51522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4789 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)(4,8)(5,6)(17,18,23,21)(19,22,25,31)(20,28,30,24)(26,32,27,29);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,25)(18,22,21,31)(20,27,30,26)(24,32,28,29);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,27)(31,32);(2,6)(4,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,18)(19,22,25,31)(20,28,30,24)(26,29,27,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,28)(22,32)(23,30)(25,26)(29,31);(1,4,5,7)(2,8,6,3)(9,10)(11,13)(12,15)(14,16)(17,22)(18,25)(19,21)(20,29)(23,31)(24,27)(26,28)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8565151);