#368747

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1725 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)(5,6,10,14,11,15,9,23)(7,12,18,29,17,27,24,28)(8,19,22,31,21,13,25,32)(16,26,34,30,33,36,20,35);(2,4)(5,7,10,18,11,17,9,24)(6,12,14,29,15,27,23,28)(8,20,22,16,21,34,25,33)(13,30,32,36,19,35,31,26);(5,8,9,25,11,21,10,22)(6,13,14,32,15,19,23,31)(7,16,24,20,17,33,18,34)(12,26,29,30,27,36,28,35);(2,4)(8,21)(13,19)(16,33)(20,34)(22,25)(26,36)(30,35)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=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, 275943);