#7572636

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7479832);