#427877

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,24)(18,27)(19,21)(23,31)(25,30)(26,28)(29,35)(32,34)(33,36);(5,7)(6,12)(8,18)(9,26)(10,20)(11,21)(13,29)(14,19)(15,17)(16,28)(22,32)(23,36)(24,34)(25,35)(27,31)(30,33);(5,8,11,25)(6,13,16,31)(7,18,21,35)(9,24,10,23)(12,27,28,29)(14,22,15,30)(17,32,19,33)(20,36,26,34);(2,4)(5,6)(7,19,21,17)(8,13)(9,14)(10,15)(11,16)(12,26,28,20)(18,33,35,32)(22,24)(23,30)(25,31)(27,34,29,36)

Polycyclic presentation:

2^9|c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,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, 335073);