#7195659

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 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)(13,14,18,23,19,24,34,41)(15,25,28,36,29,40,30,20)(16,21,31,35,27,37,32,43)(17,22,26,39,33,38,42,44);(6,9)(7,11)(13,15,19,29)(14,20,24,36)(16,26,27,42)(17,32,33,31)(18,30,34,28)(21,22,37,38)(23,40,41,25)(35,44,43,39);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,16)(14,21)(15,26)(17,30)(18,31)(19,27)(20,22)(23,35)(24,37)(25,39)(28,33)(29,42)(32,34)(36,38)(40,44)(41,43);(6,9)(10,12)(13,17,18,26,19,33,34,42)(14,22,23,39,24,38,41,44)(15,27,28,32,29,16,30,31)(20,35,25,37,36,43,40,21);(5,7,8,11)(6,10,9,12)(16,27)(17,33)(21,37)(22,38)(26,42)(31,32)(35,43)(39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,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=dh,e^c=ei,e^d=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, 7102855);