#299524

order	512
nilpotency class	3
isoclinism	isoclinic class of #299505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60817	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

996 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,g^b=gh,d^c=dhi,g^c=gh

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, 206720);