#254340

order	512
nilpotency class	3
isoclinism	isoclinic class of #254323
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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

799 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161536);