#361535

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268731);