#7534568

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,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	#72257261, #72257535, #72264068
maximal normal subgroup of	#72257261, #72257535, #72264068
maximal quotient of	#72264666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7441764);