#7182095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089291);