#128317

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
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	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#323
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 554 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)(7,15)(8,13)(9,14)(10,12)(11,16)(17,22)(18,28)(19,24)(20,26)(21,25)(23,30)(27,29)(31,33)(32,34)(35,36);(5,7,8,17,9,18,20,31,10,16,19,29,21,30,32,36)(6,11,13,22,14,23,26,33,12,15,24,27,25,28,34,35);(1,3,4,2)(6,12)(11,15)(13,24)(14,25)(22,27)(23,28)(26,34)(33,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fg,d^b=dg,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, 35513);