#128831

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36027);