#128331

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fgi,c^b=ci,d^b=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35527);