#146501

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

849 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,f^b=fhi,e^c=ei,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, 53697);