#128438

order	512
nilpotency class	3
isoclinism	isoclinic class of #105580
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285149, #72285281
maximal normal subgroup of	#72285149, #72285281
maximal quotient of	#72319502, #72319535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35634);