#253887

order	512
nilpotency class	3
isoclinism	isoclinic class of #253843
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	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

791 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161083);