#7659954

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

5030 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,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, 7567150);