#7548609

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2229 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)(19,28)(23,31)(24,32)(29,30);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32);(10,14)(11,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,25)(26,31)(27,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,25)(26,32)(27,31);(2,5)(6,8)(10,14)(13,15)(17,20,21,22)(18,25,26,27)(19,29,28,30)(23,32,31,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313531, #72313586, #72313981
maximal normal subgroup of	#72313531, #72313586, #72313981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7455805);