#7278303

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234321
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7185499);