#7620673

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(11,14)(12,13)(15,16)(19,26)(23,27)(28,32)(29,30);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,27,26,23)(20,31,25,21)(28,29,32,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,22,26)(18,23,24,27)(20,32,25,28)(21,30,31,29);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,20,21,18,22,25,31,24)(19,28,30,27,26,32,29,23);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,31)(18,20,24,25)(19,29,26,30)(23,28,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fghi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527869);