#7196836

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

1690 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)(17,18,21,26,23,27,22,32)(19,28,20,25,29,31,30,24);(3,9)(4,12)(6,13)(7,11)(17,18)(19,24)(20,31)(21,32)(22,26)(23,27)(25,29)(28,30);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19,21,20,23,29,22,30)(18,24,32,31,27,25,26,28);(3,9)(6,13)(10,15)(14,16)(17,20,22,19,23,30,21,29)(18,25,32,28,27,24,26,31);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,29)(20,30)(24,25)(28,31)

Polycyclic presentation:

2^9|a^2=fh,c^2=ghi,d^2=fi,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, 7104032);