#511862

order	512
nilpotency class	4
isoclinism	isoclinic class of #511861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2053 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 2053 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 32

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419058);