#365887

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273083);