#499237

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

918 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)(15,21)(16,22)(17,20)(18,19)(23,26)(24,27)(25,28);(1,2)(3,4)(6,9)(10,12)(13,15,16,18)(14,19,22,21)(17,24,25,26)(20,23,28,27);(2,4)(14,20)(15,18)(17,26)(19,27)(21,23)(22,28)(24,25);(5,7,8,11)(6,10,9,12)(13,15,16,18)(14,21,22,19)(17,24,25,26)(20,27,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318577, #72318584, #72318588, …
maximal normal subgroup of	#72318577, #72318584, #72318588, …
maximal quotient of	#72322452, #72322502

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406433);