#94329

order	512
nilpotency class	5
isoclinism	isoclinic class of #94311
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	#2285	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #2285 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,18,14,19,15,29)(11,20,22,37,23,38,28,34)(12,24,26,36,27,17,30,35)(16,25,31,21,33,39,32,40);(2,5,7,6)(9,11)(10,16,18,32,19,33,29,31)(12,25,27,39)(13,28)(14,23)(15,22)(17,34,35,38,24,37,36,20)(21,26,40,30)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,d^b=di,g^b=gh,e^c=ehi,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, 1525);