#401027

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1756 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)(11,21)(12,17)(13,22)(14,19)(15,18)(16,20)(23,24);(2,4)(9,11,14,22,16,12,15,23)(10,13,18,21,20,24,19,17);(2,4)(9,12)(10,17)(11,16)(13,18)(14,23)(15,22)(19,24)(20,21);(6,8)(9,13,14,21,16,24,15,17)(10,12,18,23,20,11,19,22)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=ef,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,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216148
maximal normal subgroup of	#72216148
maximal quotient of	#72215990

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308223);