#7620741

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(4,6)(5,8)(10,14)(11,12)(17,18)(19,24)(20,25)(21,23)(22,26)(27,36)(28,33)(29,37)(30,39)(31,35)(32,34)(38,43)(40,47)(41,45)(42,44)(46,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(18,23)(20,30)(22,31)(24,33)(25,38)(27,40)(29,41)(32,43)(34,39)(36,46)(42,48)(44,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,33)(25,36)(26,37)(30,40)(31,41)(32,42)(34,44)(35,45)(38,46)(39,47)(43,48);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,31,43,21,32,22,30)(18,25,35,38,23,34,26,39)(19,27,41,48,28,42,29,40)(24,36,45,46,33,44,37,47);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,23)(19,28)(20,32)(26,35)(29,41)(30,43)(36,44)(46,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=fgh,e^2=g,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527937);