#354048

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239913
maximal normal subgroup of	#72239913
maximal quotient of	#72239856, #72239858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261244);