#430508

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 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)(5,6,11,15,10,14,17,9)(7,16,21,25,20,12,27,19)(8,22,23,30,24,31,26,13)(18,32,33,28,34,36,35,29);(5,7,17,27,10,20,11,21)(6,12,9,25,14,16,15,19)(8,18,23,33,24,34,26,35)(13,29,22,32,30,28,31,36);(5,8)(6,13)(7,18)(9,22)(10,24)(11,26)(12,28)(14,30)(15,31)(16,29)(17,23)(19,32)(20,34)(21,35)(25,36)(27,33);(2,4)(5,9,17,14,10,15,11,6)(7,19,27,12,20,25,21,16)(8,22,23,30,24,31,26,13)(18,32,33,28,34,36,35,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337704);