#244498

order	512
nilpotency class	3
isoclinism	isoclinic class of #10690⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#401
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

543 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)(3,9)(4,12,14,7)(5,8)(6,11)(10,16,13,15)(17,18)(19,27,30,23)(20,24,32,38)(21,33)(22,26)(25,34)(28,39)(29,36,43,40)(31,44,45,37)(35,42)(41,46)(47,48);(1,3)(2,6,8,9)(4,13)(5,11)(7,15,12,16)(10,14)(17,19)(18,23)(20,28,32,42)(21,29)(22,30)(24,35,38,39)(25,36)(26,27)(31,41,45,48)(33,40)(34,43)(37,46,44,47);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20)(18,24,26,38)(19,28)(21,31)(22,32)(23,35,27,39)(25,37,33,44)(29,41)(30,42)(34,45)(36,46,40,47)(43,48);(17,21,22,34)(18,25,26,33)(19,29,30,43)(20,31,32,45)(23,36,27,40)(24,37,38,44)(28,41,42,48)(35,46,39,47)

Polycyclic presentation:

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

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, 151694);