#7623027

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623027
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3584	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,f^2=i,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gh,h^a=hi,g^c=ghi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215833, #72219676, #72219677
maximal normal subgroup of	#72215833, #72219676, #72219677
maximal quotient of	#72226789, #72226793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530223);