#471455

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(8,12)(9,22)(10,14)(11,17)(13,25)(16,32)(18,31)(19,20)(21,30)(23,27)(24,29)(26,35)(28,33)(34,36);(5,7,9,23,10,18,25,17)(6,11,22,15,14,27,13,31)(8,16,24,33,21,34,20,35)(12,28,29,36,30,26,19,32);(2,4)(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,19)(14,30)(15,32)(17,35)(18,34)(20,25)(22,29)(23,33)(27,28)(31,36);(5,6)(7,11)(8,19)(9,22)(10,14)(12,24)(13,25)(15,23)(16,32)(17,31)(18,27)(20,30)(21,29)(26,35)(28,33)(34,36)

Polycyclic presentation:

2^9|b^2=egh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378651);