#397265

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

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, 304461);