#7196832

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fh,c^2=gh,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 7104028);