#94332

order	512
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1345
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	C₂×#423	C₂²
prime radical	C₂⁴	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂×#423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 274 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) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=hi,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,h^a=hi,c^b=cg,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, 1528);