#357754

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226715, #72252095, #72252097, …
maximal normal subgroup of	#72226715, #72252095, #72252097, …
maximal quotient of	#72226706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264950);