#484544

order	512
nilpotency class	4
isoclinism	isoclinic class of #484194
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60734	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

908 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265737, #72308481, #72308493
maximal normal subgroup of	#72265737, #72308481, #72308493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391740);