#94298

order	512
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1351
commutator	C₂×C₂²⋊C₄	C₄²
Frattini, cosocle	#2280	C₂²
socle	C₂	#7250
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2280 ◁ G
LECS	G ▷ #2280 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=i,b^a=bc,c^a=cf,e^a=efg,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,e^c=eh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1494);