#7484922

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	#24444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 7392118);