#481012

order	512
nilpotency class	4
isoclinism	isoclinic class of #480872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36592	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#550 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1340 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,6,8,5)(4,7)(9,10,13,17,14,18,12,16)(11,15,19,22,20,23,21,24);(2,5,7,6);(9,11,14,20)(10,15,18,23)(12,21,13,19)(16,24,17,22);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=h,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,f^b=fgi,d^c=dh,f^e=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388208);