#7523076

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 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)(5,6,9,12)(7,11,8,10)(13,14)(15,21)(16,20)(17,29)(18,23)(19,24)(22,31)(25,35)(26,38)(27,37)(28,36)(30,39)(32,42)(33,41)(34,40)(43,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,18)(17,30)(19,28)(21,23)(22,39)(24,38)(25,32)(26,33)(27,34)(29,44)(31,43)(35,40)(36,41)(37,42);(2,4)(13,16,18,15)(14,21,23,20)(17,27,31,25)(19,28,33,26)(22,37,29,35)(24,38,41,36)(30,32,43,34)(39,40,44,42);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,31)(14,22,23,29)(15,25,16,27)(19,34,33,32)(20,35,21,37)(24,42,41,40)(26,43,28,30)(36,44,38,39);(5,8,9,7)(6,11,12,10)(14,23)(15,16)(22,29)(24,41)(25,27)(26,28)(30,43)(40,42)

Polycyclic presentation:

2^9|a^2=fi,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=cg,d^b=dh,e^b=ei,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, 7430272);