#7156148

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063344);