#371711

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

871 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278907);