#7623155

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623152
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215833, #72215864, #72215869
maximal normal subgroup of	#72215833, #72215864, #72215869
maximal quotient of	#72226790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530351);