#7524611

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
prime radical	#35715	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24)(25,30)(26,32)(27,31)(28,29);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,27)(18,23,24,31)(20,25,29,32)(21,30,28,26);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20,22,29)(18,21,24,28)(19,25,27,32)(23,30,31,26);(10,14)(13,16)(17,18)(19,25)(20,28)(21,29)(22,24)(23,30)(26,31)(27,32);(2,7)(6,8)(18,24)(20,29)(23,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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	#72256848, #72256852, #72256949
maximal normal subgroup of	#72256848, #72256852, #72256949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431807);