#7198391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#26926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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)(10,14)(12,15)(17,18)(19,26)(20,24)(21,31)(22,25)(23,29)(27,28)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,16,15);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,19,21,29,22,30,20,28)(18,23,24,26,25,27,31,32);(3,8)(6,7)(10,14)(11,16)(17,18)(19,27)(20,31)(21,24)(22,25)(23,30)(26,28)(29,32);(9,12,13,15)(10,11,14,16)(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=fgh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308819, #72308949
maximal normal subgroup of	#72308819, #72308949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105587);