#130048

order	512
nilpotency class	4
isoclinism	isoclinic class of #124718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#540	C₄
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

604 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,9,14,20,11,22,26,10,15,24,31,8,13,21,30)(7,16,19,33,34,12,27,28,17,25,23,36,18,29,35,32);(6,11)(8,20)(9,22)(12,25)(13,15)(14,31)(16,29)(18,34)(19,27)(21,24)(23,35)(28,32);(2,4)(5,7)(6,12,15,29)(8,18)(9,23)(10,17)(11,25,13,16)(14,32,31,28)(19,24)(20,34)(21,27)(22,35)(26,33,30,36)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37244);