#119653

order	512
nilpotency class	3
isoclinism	isoclinic class of #10456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,e^c=ehi,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239518, #72239519
maximal normal subgroup of	#72239518, #72239519
maximal quotient of	#72239498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26849);