#7339788

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 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)(19,27)(20,25)(21,36)(22,23)(24,32)(26,29)(28,40)(30,43)(31,37)(33,42)(34,38)(35,41)(39,45)(44,48)(46,47);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(18,23)(19,28)(25,38)(26,36)(30,44)(31,32)(41,42)(45,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19,29,31,22,28,21,32)(18,24,36,40,23,37,26,27)(20,30,33,46,34,44,35,45)(25,39,41,48,38,47,42,43);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,20,22,34)(18,25,23,38)(19,30,28,44)(21,35,29,33)(24,39,37,47)(26,42,36,41)(27,43,40,48)(31,46,32,45);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(17,21,22,29)(18,26,23,36)(19,31,28,32)(20,33,34,35)(24,40,37,27)(25,41,38,42)(30,45,44,46)(39,43,47,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fg,d^2=f,e^2=f,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=dhi,e^c=ei,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, 7246984);