#416417

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

930 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)(5,6,10,15)(7,18,20,13)(8,22)(9,26,21,29)(11,17,12,16)(14,25)(19,28)(23,36)(24,31)(27,32)(30,35)(33,34);(2,4)(5,7,11,21,10,20,12,9)(6,13,17,26,15,18,16,29)(8,23,24,34,25,35,27,19)(14,30,32,28,22,36,31,33);(2,4)(5,8)(6,14)(7,19)(9,23)(10,25)(11,27)(12,24)(13,28)(15,22)(16,32)(17,31)(18,33)(20,34)(21,35)(26,30)(29,36);(5,9,12,20,10,21,11,7)(6,13,17,26,15,18,16,29)(8,23,24,34,25,35,27,19)(14,28,31,30,22,33,32,36)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 323613);