#353607

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 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)(7,16,19,12)(8,13)(9,14)(10,15)(11,18,25,33)(17,31,20,26)(21,28)(22,29)(23,30)(24,32,27,34)(35,36);(5,7)(6,11,15,25)(8,20)(9,18)(10,19)(12,14,16,30)(13,27,29,24)(17,22)(21,34)(23,33)(26,36,31,28)(32,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,14)(13,26)(15,16)(20,22)(21,32)(23,33)(24,28)(25,30)(27,36)(29,31)(34,35);(5,8,9,21,10,22,23,35)(6,13,14,28,15,29,30,36)(7,17,18,32,19,20,33,34)(11,24,16,31,25,27,12,26)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257017, #72257160, #72257218, …
maximal normal subgroup of	#72257017, #72257160, #72257218, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260803);