#97388

order	512
nilpotency class	3
isoclinism	isoclinic class of #97372
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

900 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,f^c=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, 4584);