#372627

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 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)(9,10,16,20,15,22,14,21)(11,23,18,17,26,28,25,12)(13,29,30,19,31,35,32,36)(24,33,34,40,39,27,38,37);(2,5)(6,7)(9,11,16,18,15,26,14,25)(10,17,20,28,22,12,21,23)(13,24,30,34,31,39,32,38)(19,33,35,40,36,27,29,37);(9,12,15,17)(10,18,22,25)(11,21,26,20)(13,27,31,33)(14,23,16,28)(19,34,36,38)(24,29,39,35)(30,40,32,37);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,24)(12,27)(14,32)(15,31)(16,30)(17,33)(18,34)(20,35)(21,29)(22,36)(23,37)(25,38)(26,39)(28,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279823);