#195431

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,22,28,23,26,21,27)(19,29,33,24,31,39,32,40)(20,34,36,25,37,42,35,41)(30,38,46,47,44,48,45,43);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,40)(27,39)(28,29)(34,47)(35,46)(36,45)(37,44)(41,48)(42,43);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,44)(32,45)(33,46)(39,47)(40,48);(18,26)(19,31)(25,41)(27,28)(30,44)(32,33)(34,42)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102627);