#97432

order	512
nilpotency class	3
isoclinism	isoclinic class of #97372
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

680 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=gh,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,f^c=fi

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, 4628);