#250006

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	#3537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1001 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)(3,12)(4,14)(5,8)(6,11)(7,10)(9,16)(13,15)(17,18,22,26)(19,23,29,36)(20,24,31,38)(21,32,33,25)(27,39,41,34)(28,42,43,35)(30,37,44,45)(40,47,48,46);(1,3,7,13)(2,5,10,4)(6,14,16,8)(9,15,11,12)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,40)(31,41)(32,42)(33,43)(37,46)(38,39)(44,48)(45,47);(1,4,7,5)(2,8,10,14)(3,9,13,11)(6,15,16,12)(17,20,22,31)(18,24,26,38)(19,27,29,41)(21,30,33,44)(23,34,36,39)(25,37,32,45)(28,40,43,48)(35,46,42,47);(1,5,7,4)(2,3,10,13)(6,15,16,12)(8,9,14,11)(17,21,22,33)(18,25,26,32)(19,28,29,43)(20,30,31,44)(23,35,36,42)(24,37,38,45)(27,40,41,48)(34,46,39,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,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, 157202);