#429974

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1555 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)(5,6,10,14)(7,12)(8,20,9,13)(11,15,24,31)(16,25)(17,26)(18,28)(19,27)(21,29)(22,32,23,30)(33,35)(34,36);(5,7,10,18)(6,12)(8,21)(9,17)(11,19,24,33)(13,29,20,26)(14,28)(15,27)(16,23)(22,34)(25,30,36,32)(31,35);(5,8,10,9)(6,13,14,20)(7,16,18,34)(11,23,24,22)(12,25,28,36)(15,32,31,30)(17,33,21,19)(26,35,29,27);(2,4)(5,9,10,8)(6,13,14,20)(7,17)(11,22,24,23)(12,26)(15,32,31,30)(16,33)(18,21)(19,34)(25,35)(27,36)(28,29)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,g^d=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, 337170);