#499477

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322650
maximal normal subgroup of	#72322650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406673);