#128225

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1744
prime radical	C₂³	#431
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fg,e^b=eg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210619, #72210629, #72210635
maximal normal subgroup of	#72210619, #72210629, #72210635
maximal quotient of	#72210645, #72234277

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35421);