#238672

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	#34976	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145868);