#461619

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,12,16,10,17,11,28)(7,13,26,32,20,33,21,35)(8,22,29,31,24,36,19,14)(9,25,23,30,18,15,27,34);(5,7,10,20)(6,13,17,33)(8,23,24,27)(9,19,18,29)(11,26,12,21)(14,15,31,25)(16,35,28,32)(22,30,36,34);(1,2,3,4)(5,8,11,19,10,24,12,29)(6,14,16,22,17,31,28,36)(7,18,21,23,20,9,26,27)(13,30,32,15,33,34,35,25);(5,9,10,18)(6,15)(7,19,20,29)(8,21,24,26)(11,27,12,23)(13,31)(14,33)(16,34)(17,25)(22,35)(28,30)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368815);