#92836

order	512
nilpotency class	4
isoclinism	isoclinic class of #92835
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1270
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1744	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ #1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

171 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,d^2=g,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32);