#244495

order	512
nilpotency class	3
isoclinism	isoclinic class of #10690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#401
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#401
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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)(20,24,33,39)(21,25)(22,26)(28,37)(29,41,43,36)(31,35)(32,38,45,48)(34,40)(42,47)(44,46);(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)(24,37,39,35)(25,36)(26,27)(32,44,45,42)(34,43)(38,47,48,46)(40,41);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20)(18,24,26,39)(19,28)(21,32)(22,33)(23,35,27,37)(25,38,40,48)(29,42)(30,31)(34,45)(36,46,41,47)(43,44);(17,21,22,34)(18,25,26,40)(19,29,30,43)(20,32,33,45)(23,36,27,41)(24,38,39,48)(28,42,31,44)(35,46,37,47)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,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, 151691);