#7529238

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,33)(22,27)(23,28)(26,35)(29,39)(30,42)(31,41)(32,40)(34,43)(36,46)(37,45)(38,44)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,22)(21,34)(23,32)(25,27)(26,43)(28,42)(29,36)(30,37)(31,38)(33,48)(35,47)(39,44)(40,45)(41,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,22,19)(18,25,27,24)(21,31,35,29)(23,32,37,30)(26,41,33,39)(28,42,45,40)(34,36,47,38)(43,44,48,46);(10,14)(13,16)(17,21,22,35)(18,26,27,33)(19,29,20,31)(23,38,37,36)(24,39,25,41)(28,46,45,44)(30,47,32,34)(40,48,42,43);(1,4,5,3)(2,7,8,6)(18,27)(19,20)(26,33)(28,45)(29,31)(30,32)(34,47)(44,46)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,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, 7436434);