#7660672

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

4810 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)(10,12)(11,15)(17,18,20,22)(19,23,24,21);(9,10,13,12)(11,16,15,14)(17,19,20,24)(18,21,22,23);(1,3,7,6)(2,5,8,4)(9,11,13,15)(10,14,12,16);(1,4,7,5)(2,3,8,6)(9,12,13,10)(11,16,15,14);(1,5,7,4)(2,3,8,6);(1,6,7,3)(2,5,8,4)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,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, 7567868);