#492528

order	512
nilpotency class	4
isoclinism	isoclinic class of #492528
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₈²
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214134, C₃×#492528
maximal normal subgroup of	#72214134, C₃×#492528
maximal quotient of	#72216412, #72216414, #72226920, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399724);