#213590

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,g^c=gh

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, 120786);