#7012719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#24272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313813, #72313869, #72314010
maximal normal subgroup of	#72313813, #72313869, #72314010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6919915);