#7148453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(7,11)(10,12)(13,14,20,17)(15,24,28,40)(16,29,32,22)(18,23,19,34)(21,26,33,27)(25,42,30,41)(31,39,35,38)(36,43,37,44);(6,9)(7,11)(13,15,20,28)(14,21,17,33)(16,30,32,25)(18,27,19,26)(22,37,29,36)(23,40,34,24)(31,43,35,44)(38,42,39,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,32)(14,22,17,29)(15,25,28,30)(18,35,19,31)(21,36,33,37)(23,39,34,38)(24,41,40,42)(26,44,27,43);(6,9)(10,12)(13,17,20,14)(15,24,28,40)(16,22,32,29)(18,34,19,23)(21,26,33,27)(25,42,30,41)(31,38,35,39)(36,43,37,44);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,34)(15,26,28,27)(16,31,32,35)(21,24,33,40)(22,38,29,39)(25,43,30,44)(36,42,37,41)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055649);