#509402

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72344000
maximal normal subgroup of	#72344000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416598);