#372909

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi

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, 280105);