#347030

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
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	#62879	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1977 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,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,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254226);