#253299

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,g^b=gh,e^c=eh,f^c=fi,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, 160495);