#207278

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

870 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)(9,10,13,17,14,18,12,15)(11,16,21,24,19,23,20,22)(25,26)(27,30,34,37)(28,35)(29,31)(32,36)(33,39,40,38);(2,5)(6,7)(10,15,18,17)(16,22,23,24)(25,27,29,34)(26,30)(28,33,36,40)(31,37)(32,38)(35,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23)(26,31)(28,36)(30,37)(33,40);(11,19)(16,23)(20,21)(22,24)(25,28,29,36)(26,32,31,35)(27,33,34,40)(30,38,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fi,d^2=i,e^2=hi,g^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, 114474);