#365193

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,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, 272389);