#253378

order	512
nilpotency class	3
isoclinism	isoclinic class of #253363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18,22,26)(19,23,29,37)(20,24,30,38)(21,32,34,25)(27,40,33,35)(28,42,44,36)(31,46,45,39)(41,47,43,48);(4,13)(6,15)(7,12)(10,14)(19,27,29,33)(20,30)(23,35,37,40)(24,38)(28,43,44,41)(31,45)(36,48,42,47)(39,46);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19)(18,23)(20,28)(21,33)(22,29)(24,36)(25,40)(26,37)(27,34)(30,44)(31,43)(32,35)(38,42)(39,48)(41,45)(46,47);(17,20,22,30)(18,24,26,38)(19,28,29,44)(21,31,34,45)(23,36,37,42)(25,39,32,46)(27,41,33,43)(35,47,40,48)

Polycyclic presentation:

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

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, 160574);