#7156174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×#3287	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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)(13,14,19,20,18,24,17,23)(15,25,32,37,27,21,28,34)(16,22,33,35,31,40,29,39)(26,38,42,44,30,36,41,43);(6,9)(7,11)(14,20)(16,29)(17,19)(21,34)(23,24)(25,37)(26,41)(28,32)(30,42)(31,33)(35,39)(43,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,30)(17,28)(18,27)(19,32)(20,34)(22,38)(23,37)(24,25)(26,31)(29,42)(33,41)(35,44)(36,40)(39,43);(6,9)(10,12)(13,16,19,33,18,31,17,29)(14,22,20,35,24,40,23,39)(15,26,32,42,27,30,28,41)(21,36,34,43,25,38,37,44);(5,7,8,11)(6,10,9,12)(15,27)(21,25)(26,30)(28,32)(34,37)(36,38)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fi,e^2=h,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=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063370);