#7652882

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

5065 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:

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

Polycyclic presentation:

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

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, 7560078);