#207283

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

886 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)(3,7)(4,5)(6,8)(9,10,13,17,14,18,12,16)(11,19,22,23,20,24,21,15)(25,26)(27,30,33,36)(28,31)(29,32)(34,37,39,40)(35,38);(2,5)(6,7)(9,11)(10,15,18,23)(12,21)(13,22)(14,20)(16,24,17,19)(25,27)(26,30,32,36)(28,34)(29,33)(31,37,38,40)(35,39);(1,3,4,8)(2,6,5,7)(11,20)(15,23)(19,24)(21,22)(27,33)(28,35)(30,36)(31,38);(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,24,23,19)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi,d^c=di

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, 114479);