#495482

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

547 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)(7,19)(8,18)(9,16)(10,25)(11,17)(12,14)(13,21)(15,24)(20,34)(22,29)(23,31)(26,33)(27,32)(28,35)(30,36);(1,3,4,2)(5,7)(6,13)(8,20)(9,21)(10,26)(11,22)(12,27)(14,28)(15,29)(16,32)(17,30)(18,33)(19,25)(23,36)(24,35)(31,34);(5,8,11,12)(6,14,17,18)(7,20,22,27)(9,24,23,10)(13,28,30,33)(15,16,25,31)(19,34,29,32)(21,35,36,26);(5,9,8,10,11,23,12,24)(6,15,14,31,17,25,18,16)(7,21,20,26,22,36,27,35)(13,29,28,34,30,19,33,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239798, #72239799, #72239840
maximal normal subgroup of	#72239798, #72239799, #72239840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402678);