#97159

order	512
nilpotency class	3
isoclinism	isoclinic class of #97096
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

817 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4355);