#454979

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1453 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)(5,6,11,16,10,17,12,27)(7,13,21,29,20,30,22,33)(8,15,24,32,9,26,25,14)(18,31,19,35,34,36,23,28);(5,7,10,20)(6,13,17,30)(8,23,9,19)(11,22,12,21)(14,31,32,36)(15,28,26,35)(16,33,27,29)(18,24,34,25);(2,4)(5,8)(6,14)(7,18)(9,10)(11,25)(12,24)(13,28)(15,27)(16,26)(17,32)(19,22)(20,34)(21,23)(29,36)(30,35)(31,33);(5,9,10,8)(6,15)(7,19,20,23)(11,25,12,24)(13,28)(14,27)(16,32)(17,26)(18,21,34,22)(29,31)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 362175);