#7631059

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

5284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(2,5)(3,8)(10,14)(11,16)(17,18,20,22)(19,21,23,24);(1,2,6,5)(3,4,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24);(1,3,6,8)(2,7,5,4)(9,11,13,16)(10,15,14,12);(3,8)(4,7);(1,4)(2,3)(5,8)(6,7);(1,5,6,2)(3,4,8,7)(9,12,13,15)(10,16,14,11)(17,19,20,23)(18,21,22,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,f^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, 7538255);