#245924

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
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	#18993	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi,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, 153120);