#420423

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1637 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)(6,12,15,24)(7,16,19,25)(14,20,29,35)(18,30,33,36);(5,6)(7,17)(8,20)(9,24)(10,12)(11,15)(13,19)(14,21)(16,26)(18,32)(22,29)(23,35)(25,28)(27,36)(30,31)(33,34);(5,7,9,16,11,19,10,25)(6,13,12,26,15,17,24,28)(8,18,21,30,23,33,22,36)(14,27,20,34,29,31,35,32);(2,4)(5,8,9,21,11,23,10,22)(6,14,12,20,15,29,24,35)(7,18,16,30,19,33,25,36)(13,27,26,34,17,31,28,32)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^c=fi,g^c=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, 327619);