#7586605

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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7493801);