#413511

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250141, #72250145, #72250174, …
maximal normal subgroup of	#72250141, #72250145, #72250174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320707);