#147602

order	512
nilpotency class	4
isoclinism	isoclinic class of #129431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

931 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 931 conjugacy classes of subgroups — too many to draw. Only its 138 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)(7,17)(8,21,9,14)(10,15)(11,16)(12,18,29,23)(13,22,32,24)(19,30)(20,33)(25,27)(26,31)(28,35,34,36);(1,2)(3,4)(5,7,11,20)(6,12,16,29)(8,18,9,23)(10,25,26,19)(13,30,32,27)(14,33,21,17)(15,34,31,28)(22,36,24,35);(2,4)(5,8,11,9)(6,13)(7,18,20,23)(10,22,26,24)(12,27)(14,31)(15,21)(16,32)(17,28)(19,35,25,36)(29,30)(33,34)

Polycyclic presentation:

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

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, 54798);