#476001

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 660 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 — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214515, #72216230, #72216750, …
maximal normal subgroup of	#72214515, #72216230, #72216750, …
maximal quotient of	#72214172, #72268633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383197);