#504285

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,8,9)(7,11,12,10)(13,14,20,24)(15,25,21,18)(16,26,22,17)(19,27,28,23);(1,2)(3,4)(6,9)(10,11)(13,15,18,14,19,22,17,23,20,21,25,24,28,16,26,27);(2,4)(13,14,17,21,28,27,18,22,20,24,26,15,19,23,25,16);(5,7,8,12)(6,10,9,11)(13,16,18,27,19,15,17,14,20,22,25,23,28,21,26,24)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=ehi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319014, #72319556, #72319574
maximal normal subgroup of	#72319014, #72319556, #72319574
maximal quotient of	#72322595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411481);