#7712683

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	#37450	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619879);