#250055

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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	C₂×C₄²	C₂⁴
prime radical	#61788	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 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)(4,12)(5,10)(6,9)(7,14)(8,16)(13,15)(17,18,22,26)(19,23,28,38)(20,30,31,42)(21,33,36,25)(24,35,40,32)(27,43,44,47)(29,41,34,39)(37,46,48,45);(1,3)(2,4)(5,7)(6,13)(8,15)(9,14)(10,11)(12,16)(17,19,22,28)(18,23,26,38)(20,27,31,44)(21,34,36,29)(24,37,40,48)(25,41,33,39)(30,47,42,43)(32,46,35,45);(1,4)(2,7)(3,8)(5,11)(6,14)(9,12)(10,13)(15,16)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,42)(26,40)(28,44)(29,46)(30,33)(32,36)(34,45)(38,48)(39,47)(41,43);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,12)(11,16)(19,28)(21,36)(23,38)(25,33)(27,44)(30,42)(32,35)(37,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157251);