#237358

order	512
nilpotency class	3
isoclinism	isoclinic class of #237345
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60917	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1141 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)(3,7,10,16)(4,12,14,8)(6,13,15,11)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,34)(29,42)(30,39)(31,41)(33,38)(35,43)(36,44)(37,40)(45,48)(46,47);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,42)(26,39)(27,40)(28,41)(29,36)(32,47)(34,45)(35,46)(38,43)(44,48);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,32)(27,43)(28,44)(30,46)(31,45)(33,37)(39,42)(40,48)(41,47);(3,10)(6,15)(7,16)(11,13)(17,21,23,22)(18,26,28,27)(19,30,31,37)(20,34,35,36)(24,39,41,40)(25,43,44,32)(29,33,45,46)(38,48,47,42)

Polycyclic presentation:

2^9|a^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,d^b=dh,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144554);