#7196816

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
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

1713 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)(17,18,23,32,24,27,22,28)(19,21,26,25,29,31,30,20);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,27)(21,32)(22,26)(23,30)(24,29)(28,31);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,31,24,25,23,21)(18,26,32,29,27,30,28,19);(3,9)(6,13)(10,15)(14,16)(17,21,22,20,24,31,23,25)(18,26,28,19,27,30,32,29);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,29)(26,30)(28,32)

Polycyclic presentation:

2^9|a^2=fh,c^2=gh,d^2=f,e^2=hi,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104012);