#482801

order	512
nilpotency class	4
isoclinism	isoclinic class of #482722
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×C₂²⋊C₄	C₂⁴
prime radical	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1207 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1207 conjugacy classes of subgroups — too many to draw. Only its 198 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)(10,15,17,21)(16,22,24,23);(2,5)(3,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22);(9,11,14,19)(10,16,17,24)(12,20,13,18)(15,22,21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,e^2=h,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=dhi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389997);