#141098

order	512
nilpotency class	4
isoclinism	isoclinic class of #125425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

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

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, 48294);