#251981

order	512
nilpotency class	3
isoclinism	isoclinic class of #251967
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,9)(4,13)(5,8)(6,12)(7,15)(10,14)(11,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,16,14)(13,15)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,3)(2,6)(4,14)(5,11)(7,16)(8,10)(9,13)(12,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30);(4,15)(7,13)(9,16)(12,14)(20,27)(23,31)(25,32)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159177);