#455157

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 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)(5,6,11,15)(7,16,14,10)(8,19,23,12)(9,24,27,13)(17,28,20,22)(18,26,31,32)(21,33,35,29)(25,30,36,34);(5,7,10,6,11,14,16,15)(8,20,22,12,23,17,28,19)(9,18,26,13,27,31,32,24)(21,34,25,29,35,30,36,33);(2,4)(5,8)(6,12)(7,17)(9,25)(10,28)(11,23)(13,30)(14,20)(15,19)(16,22)(18,33)(21,26)(24,34)(27,36)(29,31)(32,35);(5,9,10,26,11,27,16,32)(6,13,14,31,15,24,7,18)(8,21,22,25,23,35,28,36)(12,29,17,30,19,33,20,34)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,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, 362353);