#488927

order	512
nilpotency class	4
isoclinism	isoclinic class of #12364
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1629
prime radical	#61105	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

913 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 913 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214383, #72214756, #72214826, …
maximal normal subgroup of	#72214383, #72214756, #72214826, …
maximal quotient of	#72224799, C₄×#12364

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396123);