#94274

order	512
nilpotency class	5
isoclinism	isoclinic class of #94180
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1393
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#124	C₂²
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₄×#124 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=fgh,e^2=fghi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,d^b=dg,d^c=dhi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1470);