#475913

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	#2679
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

864 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,16)(13,26)(14,18)(15,22)(17,27)(20,34)(21,35)(23,32)(24,31)(25,33)(28,39)(29,38)(30,36)(37,40);(9,11)(10,15,18,29)(12,20)(13,22)(14,21)(16,30,33,28)(17,19,26,35)(23,39)(24,36)(25,37)(27,38)(31,40,32,34);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,20)(13,23)(14,25)(15,28)(17,31)(18,33)(19,34)(21,37)(22,39)(24,27)(26,32)(29,30)(35,40)(36,38);(11,21)(12,23,25,24)(15,29)(16,31,33,32)(19,35)(20,36,37,39)(22,38)(28,40,30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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

derived subgroup of	#72268620, #72268621, #72268622
Fitting subgroup of	#72268620
maximal subgroup of	#72268618, #72268620
maximal normal subgroup of	#72268618, #72268620
maximal quotient of	#72268630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383109);