#241117

order	512
nilpotency class	3
isoclinism	isoclinic class of #241114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148313);