#502085

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33517	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,13,11,16,8,14)(7,12,25,30,20,29,18,27)(9,15,23,31,24,34,21,32)(17,35,36,26,22,33,19,28);(2,4)(5,7,8,18,11,20,10,25)(6,12,14,27,16,29,13,30)(9,22,21,36,24,17,23,19)(15,33,32,26,34,35,31,28);(6,13)(7,17)(8,10)(9,21)(12,26)(14,16)(18,36)(19,25)(20,22)(23,24)(27,33)(28,29)(30,35)(31,32);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250979, #72252778, #72252788
maximal normal subgroup of	#72250979, #72252778, #72252788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409281);