#244497

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

Polycyclic presentation:

2^9|a^2=efg,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, 151693);