#7518535

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(7,11)(8,9)(10,12)(13,14,17,21)(15,23,25,19)(16,27,29,20)(18,22,31,38)(24,34,42,41)(26,39,32,35)(28,36,43,44)(30,40,33,37);(1,2)(3,4)(6,9)(10,11)(13,15,17,25)(14,19,21,23)(16,28,29,43)(18,32,31,26)(20,36,27,44)(22,39,38,35)(24,33,42,30)(34,40,41,37);(2,4)(13,15)(14,19)(16,24)(17,25)(18,26)(20,34)(21,23)(22,35)(27,41)(28,30)(29,42)(31,32)(33,43)(36,37)(38,39)(40,44);(2,4)(13,16,17,29)(14,20,21,27)(15,24,25,42)(18,33,31,30)(19,34,23,41)(22,40,38,37)(26,43,32,28)(35,44,39,36);(5,7,8,12)(6,10,9,11)(14,21)(15,25)(20,27)(22,38)(24,42)(26,32)(28,43)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,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, 7425731);