#7522412

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	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2206 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,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, 7429608);