#362021

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,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, 269217);