#403007

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1232 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)(3,4)(5,6,10,25,9,15,11,16)(7,12,19,27,20,28,18,13)(8,21,23,14,22,35,24,30)(17,29,32,26,33,36,31,34);(5,7,11,18,9,20,10,19)(6,12,25,27,15,28,16,13)(8,17,24,31,22,33,23,32)(14,26,35,36,30,34,21,29);(5,7)(6,13)(8,17)(9,20)(10,19)(11,18)(12,25)(14,29)(15,27)(16,28)(21,34)(22,33)(23,32)(24,31)(26,35)(30,36);(2,4)(5,8,11,24,9,22,10,23)(6,14,25,35,15,30,16,21)(7,17,18,31,20,33,19,32)(12,26,27,36,28,34,13,29)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 310203);