#7631057

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4898 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)(10,14)(15,16)(17,18,23,20)(19,22,26,24)(21,25,31,28)(27,30,29,32);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,15,14,16)(18,20)(19,26)(25,28)(27,29);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,16,14,15)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32);(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=g,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, 7538253);