#138217

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45413);