#127745

order	512
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34941);