#241620

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,18,21,15)(12,16,14,22)(19,23,20,24)(25,26,31,34)(27,32,38,47)(28,39,43,52)(29,44,30,35)(33,42,51,41)(36,53,54,50)(37,49,45,48)(40,46,55,56);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,18)(22,24)(25,27,31,38)(26,32,34,47)(28,40,43,55)(29,37,30,45)(33,50,51,53)(35,48,44,49)(36,41,54,42)(39,46,52,56);(11,19)(12,14)(15,23)(16,22)(18,24)(20,21)(25,28,31,43)(26,33,34,51)(27,36,38,54)(29,42,30,41)(32,46,47,56)(35,52,44,39)(37,55,45,40)(48,50,49,53);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(32,47)(33,51)(35,44)(39,52)(46,56)(48,49)(50,53)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=ghi

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, 148816);