#237319

order	512
nilpotency class	3
isoclinism	isoclinic class of #237305
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	#60965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 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)(3,11)(4,8)(5,14)(6,10)(7,12)(9,15)(13,16)(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)(2,7)(4,6)(5,13)(8,9)(10,16)(11,14)(12,15)(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);(1,4,6,3)(2,8,10,11)(5,13,15,12)(7,14,16,9)(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);(3,4)(7,16)(8,11)(12,13)(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|c^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=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144515);