#508699

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415895);