#7631253

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4988 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538449);