#397111

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 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)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,30)(20,29)(21,28)(22,35)(23,27)(24,32)(25,31)(26,33)(34,36);(5,7,12,21,11,20,10,9)(6,13,18,29,17,28,16,15)(8,22,24,19,26,23,25,34)(14,30,31,35,33,36,32,27);(5,8,10,24,11,26,12,25)(6,14,16,31,17,33,18,32)(7,19,9,23,20,34,21,22)(13,27,15,30,28,35,29,36);(2,4)(5,9,10,20,11,21,12,7)(6,15,16,28,17,29,18,13)(8,23,24,34,26,22,25,19)(14,30,31,35,33,36,32,27)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,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, 304307);