#353346

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	C₂⁵	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4166 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)(7,11,17,24)(8,18,20,12)(9,13)(10,14)(15,27)(16,23,30,34)(19,31,32,25)(21,26)(22,29)(28,35)(33,36);(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,22,18,27)(13,23,26,34)(19,28)(20,29)(21,30)(25,33,31,35)(32,36);(2,4)(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,25)(14,18)(16,28)(17,29)(21,32)(23,33)(24,27)(26,31)(30,36)(34,35);(5,9,10,21)(6,13,14,26)(7,16,17,30)(8,19,20,32)(11,23,24,34)(12,25,18,31)(15,28,29,36)(22,33,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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	#72237659, #72237662, #72237666, …
maximal normal subgroup of	#72237659, #72237662, #72237666, …
maximal quotient of	#72303630, #72312125, #72312160, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260542);