#110410

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17606);