#7195649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2420 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)(15,25,33,38,27,21,28,35)(16,29,34,40,32,22,30,36)(26,37,42,43,31,39,41,44);(6,9)(7,11)(14,20)(16,30)(17,19)(21,35)(23,24)(25,38)(26,41)(28,33)(31,42)(32,34)(36,40)(43,44);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,18,27)(14,21,24,25)(16,31,32,26)(17,28,19,33)(20,35,23,38)(22,39,29,37)(30,42,34,41)(36,44,40,43);(6,9)(10,12)(13,16,17,30,18,32,19,34)(14,22,23,40,24,29,20,36)(15,26,28,41,27,31,33,42)(21,37,38,44,25,39,35,43);(5,7,8,11)(6,10,9,12)(15,27)(21,25)(26,31)(28,33)(35,38)(37,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102845);