#499481

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 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)(7,10)(8,9)(11,12)(13,14,16,20,23,27,28,25,15,18,22,26,17,21,24,19);(1,2)(3,4)(6,9)(10,12)(14,18)(19,25)(20,26)(21,27);(2,4)(14,19)(16,24)(17,23)(18,25)(20,21)(22,28)(26,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=f,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320185, #72320199, #72320231
maximal normal subgroup of	#72320185, #72320199, #72320231
maximal quotient of	#72320516, #72320895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406677);