#276361

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29794	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 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)(4,5)(6,8)(9,10,15,18,14,19,13,20)(11,21,22,24,12,16,23,17)(25,26,29,33,30,34,28,32)(27,31,35,38,37,40,36,39);(2,5)(6,7)(9,11)(10,16)(12,14)(13,23)(15,22)(17,18)(19,21)(20,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,38)(34,40);(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18)(26,32)(27,35)(28,29)(33,34)(36,37)(38,39);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)(25,28,30,29)(26,32,34,33)(27,35,37,36)(31,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183557);