#353359

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313066, #72313078, #72313137, …
maximal normal subgroup of	#72313066, #72313078, #72313137, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260555);