#92952

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	Q₈⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

178 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,7,5,10,15,16,6,11,3,12,13,9,14,8)(17,18,21,24,23,26,31,19,20,25,30,29,32,27,22,28);(2,7,9,8,11,12,10,16)(3,4)(14,15)(17,19,22,27,32,24,30,25,20,28,31,26,23,29,21,18)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,c^b=ci,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148);