#413071

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60643	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320267);