#249991

order	512
nilpotency class	3
isoclinism	isoclinic class of #249943
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61621	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1151 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)(3,6,11,8)(4,7,12,15)(10,14,16,13)(17,18)(19,29)(20,34)(21,28)(22,27)(23,32)(24,36)(25,35)(26,39)(30,45)(31,40)(33,42)(37,44)(38,41)(43,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,33)(18,23,27,41)(20,31,36,46)(21,38,39,32)(24,43,34,45)(25,40,44,48)(26,42,28,29)(30,35,47,37);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20)(18,24)(19,30)(21,35)(22,36)(23,40)(25,26)(27,34)(28,44)(29,43)(31,38)(32,46)(33,47)(37,39)(41,48)(42,45);(1,3)(2,8)(4,10)(5,11)(6,9)(7,14)(12,16)(13,15)(17,20)(18,25)(19,31)(21,37)(22,36)(23,40)(24,26)(27,44)(28,34)(29,45)(30,38)(32,47)(33,46)(35,39)(41,48)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157187);