#94080

order	512
nilpotency class	5
isoclinism	isoclinic class of #7349
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂⁴	C₄×C₈
Frattini, cosocle	C₂×#407	C₂²
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂×#407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,g^b=gi,d^c=dh,f^d=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, 1276);