#368975

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=hi

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, 276171);