#94432

order	512
nilpotency class	5
isoclinism	isoclinic class of #94431
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂²×D₄	C₄²
Frattini, prime radical, cosocle	C₂²×#127	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dfi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1628);