#141301

order	512
nilpotency class	4
isoclinism	isoclinic class of #141300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9743
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33717	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=gh,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214066, #72214153, #72214332
maximal normal subgroup of	#72214066, #72214153, #72214332
maximal quotient of	#72216424, #72227225, #72227226

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48497);