#152256

order	512
nilpotency class	5
isoclinism	isoclinic class of #149797
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2328 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 68 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)(7,11)(8,19,21,12)(9,14,24,13)(10,15)(16,31,32,36)(17,27)(18,26,33,30)(20,35,34,25)(22,28,23,29);(1,2,3,4)(5,7)(6,11)(8,20)(9,18)(10,17)(12,25)(13,30)(14,26)(15,27)(16,22)(19,35)(21,34)(23,32)(24,33)(28,31)(29,36);(5,8,10,22)(6,12,15,29)(7,16,17,20)(9,23,24,21)(11,25,27,36)(13,28,14,19)(18,34,33,32)(26,35,30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59452);