#237208

order	512
nilpotency class	3
isoclinism	isoclinic class of #10176
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1231 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fi,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, 144404);