#366603

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2284 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)(3,4)(5,6,7,8)(9,10)(11,15)(12,25)(13,17)(14,19)(16,28)(18,20)(21,36)(22,31)(23,34)(24,32)(26,39)(27,33)(29,35)(30,37)(38,40);(2,4)(9,11,20,23,14,24,13,22)(10,15,18,34,19,32,17,31)(12,26,27,38,29,21,28,37)(16,30,25,39,33,40,35,36);(6,8)(9,12)(10,16)(11,21)(13,27)(14,29)(15,30)(17,25)(18,35)(19,33)(20,28)(22,37)(23,38)(24,26)(31,39)(32,40)(34,36);(2,4)(9,13,14,20)(10,17,19,18)(11,22,24,23)(12,27,29,28)(15,31,32,34)(16,25,33,35)(21,37,26,38)(30,39,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273799);