#93137

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#123	C₂²
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,e^2=hi,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,g^a=gi,c^b=cg,d^b=dfhi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268762
maximal normal subgroup of	#72268762
maximal quotient of	#72268765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333);