#396355

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216158
maximal normal subgroup of	#72216158
maximal quotient of	#72215990

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303551);