#493106

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

770 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)(6,12,14,16)(9,23)(13,26,29,17)(15,32,31,27)(20,24)(22,25)(28,35)(30,34,33,36);(2,4)(5,6,9,15,10,16,25,32,11,14,23,31,8,12,22,27)(7,17,20,34,21,29,35,33,18,26,24,36,19,13,28,30);(5,7,8,19,11,18,10,21)(6,13,16,17,14,29,12,26)(9,24,22,35,23,20,25,28)(15,33,32,36,31,30,27,34);(7,18)(13,29)(17,26)(19,21)(20,24)(28,35)(30,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237211, #72239196, #72239465, …
maximal normal subgroup of	#72237211, #72239196, #72239465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400302);