#213426

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 258 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)(3,8)(4,6)(5,7)(10,13)(11,15)(14,16)(17,18,22,25,21,24,30,20)(19,26,29,28,27,23,31,32);(1,3,5,4)(2,6,7,8)(17,19,21,27)(18,23,24,26)(20,28,25,32)(22,29,30,31);(9,10,11,13,12,14,15,16)(19,27)(23,26)(28,32)(29,31);(1,4,5,3)(2,6,7,8)(17,20,30,24,21,25,22,18)(19,28,31,26,27,32,29,23)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120622);