#100153

order	512
nilpotency class	3
isoclinism	isoclinic class of #100139
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 624 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=ci,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,f^c=fgh

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, 7349);