#432178

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

1387 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)(6,11,14,21)(12,25,27,24)(13,18,28,33)(26,29,36,32);(5,6,9,21,10,14,23,11)(7,12,16,24,17,27,22,25)(8,18,19,13,20,33,35,28)(15,29,30,26,31,32,34,36);(5,7,10,17)(6,12)(8,15,20,31)(9,22,23,16)(11,24)(13,26)(14,27)(18,32)(19,34,35,30)(21,25)(28,36)(29,33);(2,4)(5,8)(6,13,14,28)(7,15)(9,19)(10,20)(11,18,21,33)(12,26,27,36)(16,30)(17,31)(22,34)(23,35)(24,32,25,29)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 339374);