#294426

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

826 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)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23)(25,26,29,33)(27,35,32,31)(28,37,39,36)(30,40,38,34);(2,5)(6,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,17)(18,21)(23,24)(25,27,29,32)(26,31,33,35)(28,34,39,40)(30,37,38,36);(9,12)(10,15)(11,18,19,20)(13,21)(14,22,17,24)(16,23)(25,28,38,31,29,39,30,35)(26,32,40,36,33,27,34,37);(1,3,4,8)(2,6,5,7)(10,16)(14,17)(15,23)(22,24)(26,33)(27,32)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fh,d^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201622);