#207755

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

845 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)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(9,11,13,18)(10,14)(12,17,19,23)(15,20)(16,21)(22,24)(25,27,28,34,29,33,35,39)(26,30,31,37,32,36,38,40);(2,5)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24);(1,3,4,7)(2,6,5,8)(12,19)(15,22)(17,23)(20,24)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=eh,g^b=gh,d^c=dh

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, 114951);