#353349

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4086 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)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,23,29)(8,20,21,30,22,13,24,28)(17,31,34,35,33,25,32,36);(5,7)(6,12,15,27)(8,17)(9,18)(10,23)(11,19)(13,25,20,31)(14,26,16,29)(21,32)(22,33)(24,34)(28,35,30,36);(2,4)(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,28)(15,20)(16,30)(18,32)(19,33)(23,34)(26,35)(27,31)(29,36);(5,9,11,10)(6,14,15,16)(7,18,19,23)(8,21,22,24)(12,26,27,29)(13,28,20,30)(17,32,33,34)(25,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303105, #72303109, #72303122, …
maximal normal subgroup of	#72303105, #72303109, #72303122, …
maximal quotient of	#72303591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260545);