#362300

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

840 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)(8,22,26,32)(9,12)(10,16)(11,17)(13,25,33,35)(14,21)(15,19)(20,28,23,30)(24,31,36,34)(27,29);(1,2,3,4)(5,7,19,27,11,21,10,9)(6,12,16,14,17,29,15,18)(8,23,35,24,26,20,25,36)(13,30,22,34,33,28,32,31);(2,4)(5,8,11,26)(6,13)(7,20,21,23)(9,24,27,36)(10,25,19,35)(12,28)(14,31)(15,32)(16,22)(17,33)(18,34)(29,30);(5,9,10,21,11,27,19,7)(6,14,15,12,17,18,16,29)(8,24,25,23,26,36,35,20)(13,31,32,28,33,34,22,30)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,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, 269496);