#7645354

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5090 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:

(1,2,5,7)(3,8,6,4)(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)(2,4)(5,6)(7,8)(9,11,13,15)(10,14,12,16);(1,4,5,8)(2,6,7,3)(9,12,13,10)(11,16,15,14);(1,4)(2,3)(5,8)(6,7);(1,3)(2,4)(5,6)(7,8)

Polycyclic presentation:

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

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, 7552550);