#424319

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331515);