#439380

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279655
maximal normal subgroup of	#72279655

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346576);