#207931

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=fh,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115127);