#128347

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fgi,d^b=dg,e^b=ei,g^b=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35543);