#105562

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12758);