#7241979

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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)(3,6)(4,7)(5,8)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(18,23)(22,28)(24,30)(29,32);(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,24,25,30)(20,28,27,22)(26,32,31,29);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,19,21,25)(18,24,23,30)(20,26,27,31)(22,29,28,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(18,23)(19,25)(22,28)(26,31)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7149175);