#244108

order	512
nilpotency class	3
isoclinism	isoclinic class of #238028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

454 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)(3,7)(4,13)(5,8)(6,9)(10,14)(11,15)(12,16)(17,18,21,25,22,26,28,37)(19,27,30,39,31,42,36,23)(20,32,35,46,34,24,40,41)(29,43,44,45,33,47,48,38);(1,3)(2,7)(4,11)(5,14)(6,12)(8,10)(9,16)(13,15)(17,19,28,30,22,31,21,36)(18,23,37,27,26,39,25,42)(20,33,40,48,34,29,35,44)(24,38,46,43,32,45,41,47);(1,2)(3,10)(4,8)(5,13)(6,9)(7,14)(11,16)(12,15)(17,20,21,35,22,34,28,40)(18,24,25,41,26,32,37,46)(19,29,30,44,31,33,36,48)(23,38,27,43,39,45,42,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,10,16,15)(20,34)(24,32)(29,33)(35,40)(38,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 151304);