#10521347

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

5018 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,24,22)(19,21,25,20)(23,30,31,26)(27,28,32,29);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(17,19)(18,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,25,22,19)(23,28,31,29)(26,32,30,27);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,14,13,15)(17,21,24,20)(18,25,22,19)(23,29,31,28)(26,32,30,27);(1,4,5,3)(2,6,7,8)(17,22,24,18)(19,21,25,20)(23,26,31,30)(27,28,32,29);(1,2)(3,6)(4,8)(5,7)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428543);