#7187451

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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(17,18,21,23)(19,27,30,40)(20,25,32,36)(22,34,28,26)(24,37,35,31)(29,41,43,48)(33,45,42,39)(38,47,46,44);(1,3,5,8)(2,6,4,7)(10,11)(12,13)(18,23)(20,32)(24,35)(26,34)(27,40)(29,43)(33,42)(44,47);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,13,16)(17,19,28,37,21,30,22,31)(18,24,34,40,23,35,26,27)(20,29,42,47,32,43,33,44)(25,38,45,48,36,46,39,41);(1,4,5,2)(3,6,8,7)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,29)(21,32)(22,33)(23,36)(24,38)(26,39)(27,41)(28,42)(30,43)(31,44)(34,45)(35,46)(37,47)(40,48);(9,13,14,12)(10,15,11,16)(20,32)(25,36)(29,43)(33,42)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=fg,d^2=fi,e^2=h,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, 7094647);