#7531915

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 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:

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

Polycyclic presentation:

2^9|c^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257286, #72257319, #72264354
maximal normal subgroup of	#72257286, #72257319, #72264354
maximal quotient of	#72265699

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439111);