#496253

order	512
nilpotency class	4
isoclinism	isoclinic class of #496249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21750
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3498 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 878 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)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,24,20)(8,13,26,30)(17,27,31,25)(18,32)(19,29)(21,23)(22,28);(1,3,6,15)(2,7,11,24)(4,17,19,31)(5,20,22,14)(8,16,26,29)(9,10,28,12)(13,21,30,18)(23,27,32,25);(1,4,6,19)(2,8,11,26)(3,13,15,30)(5,21,22,18)(7,23,24,32)(9,27,28,25)(10,16,12,29)(14,31,20,17);(1,3,6,15)(2,7)(4,13,19,30)(5,14,22,20)(8,23)(9,12)(10,28)(11,24)(16,25)(17,21,31,18)(26,32)(27,29)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gh,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403449);