#357818

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2722 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)(8,13)(9,14)(10,15)(11,16)(12,19)(18,29)(20,27)(21,28)(22,36)(23,30)(24,32)(25,31)(26,34)(33,35);(5,7,9,19,11,20,10,21)(6,12,15,17,16,28,14,27)(8,22,24,18,25,34,23,35)(13,29,30,36,31,33,32,26);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,30)(15,32)(16,31)(17,33)(19,22)(20,35)(21,34)(27,29)(28,36);(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,23,25,24)(12,27,28,17)(13,30,31,32)(18,22,35,34)(26,29,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,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	#72255603, #72255613, #72255620, …
maximal normal subgroup of	#72255603, #72255613, #72255620, …
maximal quotient of	#72314666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265014);