#354852

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23,25,17)(13,18)(14,20)(15,19)(21,35,36,29)(22,30)(24,37,39,31)(26,38,40,32)(27,33)(28,34);(2,5)(6,8)(9,11,14,13,15,22,28,27)(10,16,20,18,19,30,34,33)(12,21,26,24,25,36,40,39)(17,29,32,31,23,35,38,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,31)(19,23)(20,32)(22,36)(27,39)(28,40)(30,35)(33,37)(34,38);(9,13,14,22,15,27,28,11)(10,18,20,30,19,33,34,16)(12,24,26,36,25,39,40,21)(17,31,32,35,23,37,38,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224727, #72224730
maximal normal subgroup of	#72224727, #72224730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262048);