#7521164

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1788 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)(13,14,20,26)(15,21,16,22)(17,23,33,41)(18,24,19,25)(27,35,29,37)(28,36,30,38)(31,39,43,44)(32,40,34,42);(1,2)(3,4)(6,10)(11,12)(13,15,20,16)(14,21,26,22)(17,31,33,43)(18,28,19,30)(23,39,41,44)(24,36,25,38)(27,34,29,32)(35,42,37,40);(2,4)(13,16,20,15)(14,22,26,21)(17,29,33,27)(18,28,19,30)(23,37,41,35)(24,36,25,38)(31,32,43,34)(39,40,44,42);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,23)(15,27)(16,29)(18,32)(19,34)(20,33)(21,35)(22,37)(24,40)(25,42)(26,41)(28,43)(30,31)(36,44)(38,39);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,26,25)(15,28,16,30)(17,32,33,34)(21,36,22,38)(23,40,41,42)(27,43,29,31)(35,44,37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428360);