#10368053

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

4722 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)(6,8)(17,18,23,24)(19,29,31,25)(20,28,22,26)(21,27,32,30);(9,10,13,15)(11,12,16,14)(18,24)(20,22)(25,29)(27,30);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,15,12)(17,19,23,31)(18,25,24,29)(20,27,22,30)(21,26,32,28);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,20,23,22)(18,26,24,28)(19,27,31,30)(21,25,32,29);(1,3,5,7)(2,6,4,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,27)(19,28)(20,25)(22,29)(23,32)(24,30)(26,31);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,14)(15,16)(17,22,23,20)(18,28,24,26)(19,30,31,27)(21,29,32,25)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275249);