#7339447

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7246643);