#7187456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094652);