#7672527

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579723);