#7521197

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

1793 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)(8,12)(9,10)(13,14,19,25)(15,22,16,21)(17,30,33,23)(18,34,20,24)(26,36,28,39)(27,37,29,38)(31,40,43,44)(32,41,35,42);(1,2)(3,4)(6,10)(11,12)(13,15,19,16)(14,21,25,22)(17,31,33,43)(18,27,20,29)(23,40,30,44)(24,37,34,38)(26,35,28,32)(36,41,39,42);(2,4)(13,16,19,15)(14,22,25,21)(17,28,33,26)(18,27,20,29)(23,39,30,36)(24,37,34,38)(31,32,43,35)(40,42,44,41);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,23)(15,26)(16,28)(18,32)(19,33)(20,35)(21,36)(22,39)(24,41)(25,30)(27,43)(29,31)(34,42)(37,40)(38,44);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,34)(15,27,16,29)(17,32,33,35)(21,37,22,38)(23,41,30,42)(26,43,28,31)(36,40,39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,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, 7428393);