#429994

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 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)(6,12,15,24)(13,16,28,26)(14,21,29,30)(27,31,32,36);(5,6,9,24,11,15,10,12)(7,16,18,13,20,26,19,28)(8,21,22,14,23,30,25,29)(17,32,33,31,34,27,35,36);(5,7,11,20)(6,13,15,28)(8,17,23,34)(9,19,10,18)(12,26,24,16)(14,27,29,32)(21,36,30,31)(22,35,25,33);(2,4)(5,8)(6,14,15,29)(7,17)(9,22)(10,25)(11,23)(12,21,24,30)(13,27,28,32)(16,31,26,36)(18,33)(19,35)(20,34)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,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, 337190);