#499264

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

820 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281854, #72307725
maximal normal subgroup of	#72281854, #72307725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406460);