#371795

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

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

770 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:

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

Polycyclic presentation:

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

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, 278991);