#498762

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2380
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23355
prime radical	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cef,d^b=de,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405958);