#493275

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 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)(3,7)(4,6)(5,8)(9,10,13,19,14,16,27,29,15,20,26,34,12,18,25,33)(11,17,23,32,24,28,37,38,21,30,35,39,22,31,36,40);(1,3)(2,5)(4,8)(6,7)(10,16,20,18)(13,26)(17,28,30,31)(19,33,34,29)(23,35)(25,27)(32,40,39,38)(36,37);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,24)(15,21)(16,28)(18,31)(19,32)(20,30)(25,36)(26,35)(27,37)(29,38)(33,40)(34,39);(2,6)(5,7)(11,21)(17,30)(22,24)(23,35)(28,31)(32,39)(36,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239461, #72239463, #72239467, …
maximal normal subgroup of	#72239461, #72239463, #72239467, …
maximal quotient of	#72343704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400471);