#7143725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2215 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)(13,14,19,17)(15,23,26,37)(16,21,28,27)(18,31,32,22)(20,33,30,25)(24,36,38,44)(29,41,42,35)(34,43,40,39);(6,9)(7,11)(13,15,18,25,19,26,32,33)(14,20,22,23,17,30,31,37)(16,24,29,39,28,38,42,43)(21,34,35,36,27,40,41,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,28)(14,21,17,27)(15,24,26,38)(18,29,32,42)(20,34,30,40)(22,35,31,41)(23,36,37,44)(25,39,33,43);(6,9)(10,12)(13,17,19,14)(15,23,26,37)(16,27,28,21)(18,22,32,31)(20,33,30,25)(24,36,38,44)(29,35,42,41)(34,43,40,39);(5,7,8,11)(6,10,9,12)(16,28)(21,27)(24,38)(29,42)(34,40)(35,41)(36,44)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7050921);