#101222

order	512
nilpotency class	3
isoclinism	isoclinic class of #100923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

736 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=gh,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

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, 8418);