#348673

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
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	#3531	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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,d^b=dh,f^c=fi,g^c=ghi,f^d=fhi,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, 255869);