#428857

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

953 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)(5,6,11,15,10,14,26,9)(7,16,20,25,19,12,28,18)(8,21,23,30,24,31,27,13)(17,33,34,29,22,36,35,32);(5,7,10,19)(6,12,14,16)(8,22)(9,25,15,18)(11,20,26,28)(13,29)(17,24)(21,36)(23,35)(27,34)(30,32)(31,33);(5,8)(6,13)(7,17)(9,21)(10,24)(11,27)(12,29)(14,30)(15,31)(16,32)(18,33)(19,22)(20,35)(23,26)(25,36)(28,34);(2,4)(5,9,26,14,10,15,11,6)(7,18,28,12,19,25,20,16)(8,21,23,30,24,31,27,13)(17,33,34,29,22,36,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336053);