#127689

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 130 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)(7,16)(8,13)(9,22)(10,25)(11,15)(12,18)(14,23)(17,27)(19,34)(20,28)(21,30)(24,29)(26,35)(31,33)(32,36);(5,7,9,19,10,17,23,32,11,20,24,35,8,18,21,33)(6,12,14,26,13,28,22,36,15,27,30,34,25,16,29,31);(5,8,11,10)(6,13,15,25)(7,17,20,18)(9,23,24,21)(12,16,27,28)(14,29,30,22)(19,33,35,32)(26,36,34,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286539, #72286551, #72286556
maximal normal subgroup of	#72286539, #72286551, #72286556

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34885);