#146480

order	512
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1784 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1784 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53676);