#241121

order	512
nilpotency class	3
isoclinism	isoclinic class of #241114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

824 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,28)(19,29,31,44)(20,33,35,43)(21,27,22,26)(24,38,40,32)(25,36,42,37)(30,41,34,39)(45,48,47,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,23,31)(18,24,28,40)(20,34,35,30)(21,38,22,32)(25,41,42,39)(26,44,27,29)(33,47,43,45)(36,46,37,48);(4,11)(6,12)(7,14)(9,15)(17,20,21,36,23,35,22,37)(18,25,26,43,28,42,27,33)(19,30,32,46,31,34,38,48)(24,39,29,45,40,41,44,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(19,31)(20,35)(24,40)(25,42)(29,44)(32,38)(33,43)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,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, 148317);