#7409358

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7316554);