#94065

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂²×M₃₂	C₂²
socle	C₂²	#1322
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fi,d^b=dfi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1261);