#501341

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂×#2184	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

845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319120, #72319567, #72319640
maximal normal subgroup of	#72319120, #72319567, #72319640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408537);