#110183

order	512
nilpotency class	3
isoclinism	isoclinic class of #110180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 743 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,7,12,14)(5,8)(6,11)(10,15,16,13)(17,18,21,27,23,26,22,28)(19,29,32,41,31,24,35,38)(20,25,34,39,33,40,36,42)(30,37,45,47,44,43,46,48);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,22,32,23,31,21,35)(18,24,28,38,26,29,27,41)(20,30,34,46,33,44,36,45)(25,37,39,48,40,43,42,47);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,21,36,23,33,22,34)(18,25,28,39,26,40,27,42)(19,30,35,45,31,44,32,46)(24,37,38,48,29,43,41,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285395
maximal normal subgroup of	#72285395

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17379);