#493129

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239360, #72239414, #72239417, …
maximal normal subgroup of	#72239360, #72239414, #72239417, …
maximal quotient of	#72319047, #72319051, #72319980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400325);