#375086

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282282);