#117138

order	512
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,9)(8,11)(10,12)(13,14,18,22,19,24,17,23)(15,25,28,37,29,36,27,20)(16,30,33,21,34,39,32,40)(26,35,41,44,42,43,31,38);(1,2)(3,4)(13,15,17,27,19,29,18,28)(14,20,22,25,24,37,23,36)(16,31,32,42,34,41,33,26)(21,38,39,35,40,44,30,43);(2,4)(6,9)(7,12)(10,11)(13,16,19,34)(14,21,24,40)(15,26,29,42)(17,33,18,32)(20,35,37,43)(22,30,23,39)(25,38,36,44)(27,41,28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

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, 24334);