#7236192

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42673	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^2=i,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,e^c=ei,g^c=gi,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, 7143388);