#353342

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316341, #72316348, #72316351, …
maximal normal subgroup of	#72316341, #72316348, #72316351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260538);