#253880

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	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,d^2=h,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, 161076);