#244586

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1191 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1191 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151782);