#7582319

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257319, #72257477, #72257782
maximal normal subgroup of	#72257319, #72257477, #72257782
maximal quotient of	#72308715

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489515);