#7171990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313283, #72313674, #72313815
maximal normal subgroup of	#72313283, #72313674, #72313815

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7079186);