#9682693

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	#41945	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9589889);