#383695

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313971, #72313985, #72314039
maximal normal subgroup of	#72313971, #72313985, #72314039

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290891);