#7660687

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	#36434	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4891 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567883);