#7007141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2020 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,37,19,25,18,24)(15,26,32,21,28,43,29,40)(16,30,38,41,33,22,34,39)(17,35,31,42,36,23,27,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,25,40)(16,31,33,27)(17,34,36,38)(18,32,20,29)(22,35,30,23)(24,43,37,26)(39,44,41,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,33)(14,22,25,30)(15,27,28,31)(17,29,36,32)(18,34,20,38)(21,23,40,35)(24,41,37,39)(26,42,43,44);(7,12)(9,11)(13,17,20,31,19,36,18,27)(14,23,37,44,25,35,24,42)(15,16,32,38,28,33,29,34)(21,39,43,30,40,41,26,22);(6,10)(9,11)(15,28)(16,33)(21,40)(22,30)(26,43)(29,32)(34,38)(39,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914337);