#348078

order	512
nilpotency class	3
isoclinism	isoclinic class of #348070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 255274);