#493105

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

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,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	#72237211, #72239197, #72239416, …
maximal normal subgroup of	#72237211, #72239197, #72239416, …
maximal quotient of	#72319131, #72320039, C₂×#493105

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400301);