#420538

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1667 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)(6,12,15,23)(7,16,19,24)(8,20,22,25)(27,36,32,33);(5,6)(7,17)(8,21)(9,23)(10,12)(11,15)(13,19)(14,20)(16,26)(18,33)(22,29)(24,28)(25,30)(27,34)(31,32)(35,36);(2,4)(5,7,9,16,11,19,10,24)(6,13,12,26,15,17,23,28)(8,18,20,31,22,35,25,34)(14,27,21,36,30,32,29,33);(5,8,9,20,11,22,10,25)(6,14,12,21,15,30,23,29)(7,18,16,31,19,35,24,34)(13,27,26,36,17,32,28,33)

Polycyclic presentation:

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

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, 327734);