#7488848

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7396044);