#362301

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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

826 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)(8,21,24,30)(9,12)(10,16)(11,17)(13,23,31,36)(14,20)(15,26)(19,27,33,34)(22,29,35,32)(25,28);(1,2,3,4)(5,7,10,9,11,20,26,25)(6,12,15,18,17,28,16,14)(8,19,23,22,24,33,36,35)(13,27,30,32,31,34,21,29);(2,4)(5,8,11,24)(6,13)(7,19,20,33)(9,22,25,35)(10,23,26,36)(12,27)(14,29)(15,30)(16,21)(17,31)(18,32)(28,34);(5,9,10,20,11,25,26,7)(6,14,15,12,17,18,16,28)(8,22,23,33,24,35,36,19)(13,29,30,27,31,32,21,34)

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,d^b=di,h^c=hi

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, 269497);