#234403

order	512
nilpotency class	3
isoclinism	isoclinic class of #234400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141599);