#7484365

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#53096	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7391561);