#499454

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	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318459, #72318463, #72318484, …
maximal normal subgroup of	#72318459, #72318463, #72318484, …
maximal quotient of	#72322385, #72322452

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406650);