#289308

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=efh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196504);