#346498

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62870	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2151 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:

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

Polycyclic presentation:

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

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, 253694);