#299324

order	512
nilpotency class	3
isoclinism	isoclinic class of #299301
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	#60714	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

961 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 961 conjugacy classes of subgroups — too many to draw. Only its 174 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)(17,18,21,23)(19,27,29,32)(20,26,22,25)(24,28,31,30);(1,3,6,14,7,4,8,16)(2,9,11,10,12,5,13,15)(18,23)(24,31)(25,26)(27,32);(1,4,7,3)(2,10,12,15)(5,13,9,11)(6,16,8,14)(17,19,20,28,21,29,22,30)(18,24,25,32,23,31,26,27);(1,5,8,10,7,9,6,15)(2,3,13,16,12,4,11,14)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,31,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=fi,d^2=eh,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,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206520);