#356622

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312195, #72312215, #72312221
maximal normal subgroup of	#72312195, #72312215, #72312221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263818);