#427830

order	512
nilpotency class	3
isoclinism	isoclinic class of #417674
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×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(7,12)(8,19)(9,23)(10,15)(11,16)(13,22)(14,25)(17,28)(18,30)(20,33)(21,26)(24,29)(27,35)(31,36)(32,34);(5,7,11,10)(6,12,16,15)(8,17,21,22)(9,18,24,25)(13,26,28,19)(14,23,30,29)(20,34,35,31)(27,32,33,36);(5,8)(6,13)(7,17)(9,20)(10,22)(11,21)(12,19)(14,27)(15,26)(16,28)(18,31)(23,32)(24,35)(25,34)(29,36)(30,33);(2,4)(5,9,7,18,11,24,10,25)(6,14,15,29,16,30,12,23)(8,20,17,31,21,35,22,34)(13,27,26,36,28,33,19,32)

Polycyclic presentation:

2^9|b^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,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, 335026);