#7561363

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2148 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)(7,11)(8,12)(9,10)(15,23)(16,27)(19,24)(20,35)(26,38)(28,36)(29,32)(34,40);(2,3)(6,10)(11,12)(13,14,17,21)(15,24,23,19)(16,28,27,36)(18,31,30,22)(20,32,35,29)(25,41,37,44)(26,34,38,40)(33,42,39,43);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,25)(17,23)(18,26)(20,33)(21,24)(22,34)(27,37)(28,44)(29,42)(30,38)(31,40)(32,43)(35,39)(36,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16,17,27)(14,20,21,35)(15,25,23,37)(18,32,30,29)(19,33,24,39)(22,36,31,28)(26,43,38,42)(34,41,40,44);(6,10)(11,12)(15,23)(16,27)(19,24)(20,35)(26,38)(28,36)(29,32)(34,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7468559);