#97095

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#392	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

3634 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 3634 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266393
maximal normal subgroup of	#72266393
maximal quotient of	#72266397

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4291);