#7556626

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

1498 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18)(19,27)(20,32)(21,25)(22,26)(23,30)(24,33)(28,36)(29,37)(31,40)(34,45)(35,39)(38,46)(41,43)(42,48)(44,47);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,23,25,27)(20,28,33,43)(22,29,35,31)(24,36,32,41)(26,40,39,37)(34,42,46,44)(38,48,45,47);(10,14)(15,16)(17,20,21,33)(18,24,25,32)(19,28,30,43)(22,34,35,46)(23,36,27,41)(26,38,39,45)(29,42,31,44)(37,47,40,48);(9,11,13,12)(10,15,14,16)(18,25)(19,29)(22,35)(23,37)(24,32)(27,40)(28,42)(30,31)(34,46)(36,47)(41,48)(43,44);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(18,25)(20,33)(23,27)(26,39)(28,43)(34,46)(37,40)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,e^2=fi,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, 7463822);