#432007

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1244 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)(5,6,9,14,10,15,27,33)(7,16,19,30,21,29,20,11)(8,22,24,31,25,13,26,12)(17,35,18,34,23,28,36,32);(5,7,9,20,10,21,27,19)(6,11,14,29,15,30,33,16)(8,23,24,18,25,17,26,36)(12,28,22,34,31,35,13,32);(5,8)(6,12)(7,17,21,23)(9,26)(10,25)(11,28,30,35)(13,14)(15,31)(16,34,29,32)(18,19,36,20)(22,33)(24,27);(2,4)(5,8)(6,13)(7,18,21,36)(9,24)(10,25)(11,28,30,35)(12,14)(15,22)(16,32,29,34)(17,19,23,20)(26,27)(31,33)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=efi,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,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, 339203);