#100462

order	512
nilpotency class	3
isoclinism	isoclinic class of #100235
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

451 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7658);