#127692

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
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×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34888);