#7562694

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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=gi,e^2=fi,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, 7469890);