#499659

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224928, #72239796, #72239816, …
maximal normal subgroup of	#72224928, #72239796, #72239816, …
maximal quotient of	#72224927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406855);