#347180

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62873	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1931 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)(3,10,12,15)(4,8,6,13)(7,14,16,11)(18,23)(21,32)(24,34)(25,35)(26,33)(27,39)(28,41)(30,43)(36,46)(37,42)(38,44)(45,47);(1,2)(3,7)(4,8)(5,9)(6,13)(10,14)(11,15)(12,16)(17,18,22,26)(19,24,29,37)(20,25,31,38)(21,33,32,23)(27,36,39,47)(28,42,41,34)(30,44,43,35)(40,46,48,45);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,16)(10,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,34)(25,36)(26,37)(30,40)(31,39)(32,41)(33,42)(35,45)(38,47)(43,48)(44,46);(1,4,5,6)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,25)(19,27)(21,30)(22,31)(23,35)(24,36)(26,38)(28,40)(29,39)(32,43)(33,44)(34,45)(37,47)(41,48)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254376);