#348100

order	512
nilpotency class	3
isoclinism	isoclinic class of #348098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1227 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1227 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255296);