#404280

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319002, #72319049
maximal normal subgroup of	#72319002, #72319049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311476);