#92984

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#431	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

231 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,c^b=ci,d^b=dfh,d^c=dh

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, 180);