#7556620

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,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, 7463816);