#7527769

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2007 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434965);